Most common way for each pattern to form?

[hotdogPi]

Toad: I imagine it would be the pattern (not sure which one) that forms three blocks and a toad, and nearby reactions prevent the toad from being destroyed.
LWSS and MWSS: I know that a glider hitting a pre-block can form these, but I've seen several formed in other ways.
Big S: Boat + pi, or are there more common ways of forming it?

Other things I'm interested in:
What makes a long boat more common than a barge?
What percent of eaters are formed by two-glider collisions? (I imagine it's quite small.)
Are most spark coils and dead spark coils formed by two pi heptominoes exactly the right distance apart?
What makes the long snake (7 cells) and even the regular snake so rare?
How is quadpole on ship more common than tripole?

[dvgrn]

Toad: I imagine it would be the pattern (not sure which one) that forms three blocks and a toad, and nearby reactions prevent the toad from being destroyed.
LWSS and MWSS: I know that a glider hitting a pre-block can form these, but I've seen several formed in other ways.
Big S: Boat + pi, or are there more common ways of forming it?

The toad predecessor is century.

Ship: in soups with empty space around the edges, these form more often from Herschels traveling out from the boundary, than from any other cause, though a glider hitting a traffic light or blinker is also probably pretty common.

Many most-common questions like these can be answered pretty well by checking the Catagolue page for the object, and doing a survey of the immediate predecessors in the list of random C1 soup links given there.

[vivi]

Big S: Boat + pi, or are there more common ways of forming it?

i've seen a reaction between a fleet predecessor and a blinker that can form a big S too, but i have to imagine it's less common than boat + pi

[bubblegum]

How is quadpole on ship more common than tripole?

Fleet+century reaction:

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x = 11, y = 13, rule = B3/S23
8bo$9bo$9bo$6b3o2$4b2o$3bobo$3b2o$b2o$obo$2o!

(Strictly speaking those are a ship-tie and a century cousin but whatever)

[MathAndCode]

Here's something else interesting: There are two two-glider collisions that result in a loaf and blinker with the same relative position. It turns out that this is because each of those two collisions results in a traffic light predecessor placed so that its center is four cells away (orthogonally) from the nearest cell in the block. Since the block and the traffic light are each common individually, it's conceivable that some patterns could be common primarily due to arising from some combination of the two. Here is a list of what patterns arise from an interaction of a block and a traffic light predecessor by the displacement between what the center of the traffic light would be and the closest cell in the block:

• (3, 0): Depending on the orientation of the traffic light predecessor, either a block in the same place as it started or nothing
• (4, 0): Depending on the orientation of the traffic light predecessor, either a loaf and blinker, as discussed earlier, or a block and Beethoven
• (5, 0): Diehard
• {6, 0): A beehive (Although the beehive is only alone after a symmetrical sequence lasting for about 50 generations)
• (3, 1): Depending on the orientation of the traffic light predecessor, either nothing, a beehive, a traffic light displaced one cell orthogonally from where it would have formed without the block, or a constellation of one block, one beehive (which is from a honey farm that got mostly destroyed), one boat, and Herschel ash and two gliders (including the Herschel's SNG) in the same direction with a relative displacement of (52.5, 1.5)
• (4, 1): Depending on the orientation of the traffic light predecessor, either a blockade or a glider escaping a constellation of two blocks, two loaves, and three beehives, including one from a teardrop
• (5, 1): A block and a blinker
• (6, 1): Three-quarters of a traffic light (because the block and one of the blinkers mutually annihilated)
• (2, 2): Depending on the orientation of the traffic light predecessor, either nothing or a blinker
• (3, 2): Depending on the orientation of the traffic light predecessor, either nothing, a blinker, a blinker and a traffic light, or ash of the octomino
• (4, 2): Three-quarters of a traffic light (because the block and one of the blinkers mutually annihilated)
• (5, 2): Half of a traffic light (because the block and two of the blinkers mutually annihilated)
• (3, 3): Half of a traffic light (because the block and two of the blinkers mutually annihilated)
• (4, 3): A blinker, block, loaf, and toad

There are other possible interactions that would likely be common, like a traffic light predecessor interacting with a blinker or beehive, but this list already contains some interesting data, specifically concerning the formation of toads, which was one of the objects mentioned. A toad results from the interaction with a displacement of (4, 3). In addition, the octomino creates a toad, although, like with the century, it is later destroyed. In addition to an interaction between a block and a traffic light predecessor, an octomino is formed by one of the two-glider collisions, giving at least two potentially common predecessors of this toad-creating sequence.

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x = 12, y = 13, rule = TripleB3S23
2B$2B9.A$9.2A$3.B6.2A$3.B$2.B.B5$2.A$2.2A$.A.A!
#C [[ T 35 GPS 40 T 166 GPS 10 T 167 GPS 30 ]]

It's possible that the octomino sequence cut off before it could destroy the toad or the interaction between a block and a traffic light predecessor with displacement (4, 3) could account for a significant proportion of toads.
Another interesting result from looking at two-glider collisions is the fact that the pond and fishhook have two-glider syntheses despite being less common than the ship and tub, which don't. About ships, I agree with Dave Greene that the reason for ships is most likely due to the fact that they occur in Herschel ash. There are actually four two-glider collisions that make a ship, but none of them only make a ship. Instead, the ship is placed near two blocks with positioning characteristic of a Herschel, and on the other side of those two blocks lies another block (in three out of four collisions that produce a ship) or a beehive, eight blinkers (from one complete traffic light and three partially destroyed traffic lights), and two blocks. I'm not sure what the reason is for the tub, though. There's a two-glider collision that creates a tub as well as a loaf, block, and blinker (which do not have the same relative position as in the interaction between the block and traffic light predecessor with a displacement of (4, 3)), but if we count that, then we also need to count the six ponds produced from the three two-gliders collisions that result in a ∏-heptomino, so I still don't understand why the tub's greater commonness than the pond is not reflected in the ash of two-glider collisions. Granted, two-glider collisions are not mandated to have the same object frequency as 16×16 soups, but it seems unlikely that one object would appear eight times as much as a less common object.
In general, how common a particular pattern is depends more on how common its predecessors are than how simple it is (although some objects' predecessors are common because those predecessors are small). The traffic light and honey farm are common because they have small predecessors.

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x = 27, y = 21, rule = TripleB3S23
.B18.B5.B$3.A17.2A2D$.EGFA16.BCDCF$DBC.D18.FAE$24.D10$2.D$24.CE$.D.D20.DB$.CGC21.G$2.D$.ABA19.2G$22.ADBA!

Another example is that the pulsar and pentadecathalon, despite being relatively large and complex oscillators, are more common than simpler oscillators, like the clock and bipole. The reason that the pulsar is so common is because it has relatively small (and therefore likely relatively common) predecessors, like this eight-cell predecessor, which has eight eight-cell descendants that other patterns could converge to.

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x = 6, y = 4, rule = B3/S23
5bo$obo2bo$bo2bo$4b2o!

The pentadecathlon also results from a symmetric arrangement of two common traffic light predecessors.

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x = 3, y = 10, rule = B3/S23
bo2$3o5$3o2$bo!

There are probably many examples of patterns that occur much more often than their size would suggest. One particularly impressive example due to its sheer size is Lidka ash.
If you're going to attempt to determine how likely a particular object is to have formed in a particular way (which I think would provide interesting and possibly useful data), then I would also like to see results from seeing how commonly different patterns are formed, as opposed to how likely they are to form and survive. For example, the block occurs about 7.7% more frequency than the blinker according to Catalogue, but since blocks can serve as eaters, is it possible that the blinker is actually formed more often than the block but is simply less likely to survive before everything settles? Likewise, the fishhook might be overtaken by the LWSS for the same reason as well as the fact that LWSSes that aren't created near the edge of the ash (which is more likely than it may seem because LWSSes (and objects in general) are more likely to be spawned by soups that last longer before stabilizing, which tend to have larger groups of ash, so not all of the ash is near the edge) would likely crash into a still-life or oscillator instead of escaping. Also, if many toads are formed by centuries or octominoes, the toad will probably be more common as a proportion of the total number of objects, possibly even passing the long boat.

[Hdjensofjfnen]

I can at least try to answer a few of these.

What makes the long snake (7 cells) and even the regular snake so rare?

A lack of predecessors that are smaller or of comparable size. If you look closely at the soups for the snake and the python (long snake), almost all of them have to form instantaneously out of larger reactions; there's simply no small predecessors because of their density. Longer snakes, canoes, etc. are similarly rare because the long diagonal line motif needs to be created from the edge of a live reaction (edgeshot), which is difficult.

What makes a long boat more common than a barge?

Most barge appearances from soup rely on a tub predecessor getting perturbed by a stray spark. Since the tub itself is not too common due to a smaller abundance of small predecessors, barges are a bit rare. However, longboats are easily created from the edges of live reactions because they have a large variety of small, sparse, contiguous predecessors of comparable size, like this:

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x = 4, y = 5, rule = B3/S23
o$o$o$bobo$2bo!

Are most spark coils and dead spark coils formed by two pi heptominoes exactly the right distance apart?

Yep! Most of these occurrences are from symmetric areas of soups that develop without becoming disturbed, which makes the appearance of a dead or live spark coil more likely.

What percent of eaters are formed by two-glider collisions? (I imagine it's quite small.)

It's unlikely, but since the two gliders required to form the eater are 90 degrees apart from one another, it's not impossible. The way in which the two-glider collision forms the eater, however -- from two R predecessors -- is one the most common ways the eater is formed:

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x = 6, y = 5, rule = B3/S23
b2o$o$o4bo$o4bo$2b3o!

along with this small predecessor which is commonly edgeshot:

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x = 4, y = 5, rule = B3/S23
2o$2o$2bo$2bo$3bo!

[goldenratio]

Most patterns don't have a super common predecessor in which the majority of the natural occurrences are based on. For example, if you run ten claw with tail soups, you're probably going to find ten different ways that the object is produced.

The exception to this is if a pattern is a lot more common than it actually looks. The pulsar is one example. Another is 29-bit still-life #1 (the most common 29 bit still life on Catagolue which is actually so disproportionately common that it received that name.) Almost all the occurrences are caused by a pi+block+b-heptomino (or a similar sparking which results in the same immediate predecessor):

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#C Source: https://conwaylife.com/wiki/Talk:Xs29_cc0s2ticz330fgkc
x = 63, y = 22, rule = B3/S23
57b3o$56bo$56bo3b2o$55bo3b3o$27bobobobo5bobo13b3o3b2o$37bo5bo9bo2b2o2b
2o$19bo7bo23bo4bo3bo$37bo5bo7bo5b3o$19bo7bo22b2o6bo$2bo34bo5bo7b2o$b3o
6b2o3bobobobobo3bobobo7bobo7bo$2ob2o5b2o21bo9bo6b3o$19bo$33bo9bo$19bo
38b2o$33bo9bo14b2o2$27bobobo5bobobo2$49b3o$49bo2bo$49bob2o!

For the Big S, it seems to be that there are a lot of small predecessors that expand into one, here are two I found in a few minutes (converging to the same sequence at gen 2):

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x = 16, y = 4, rule = B3/S23
bo10b2o$2obobo5b2obo$3bo6bo4bo$b2o11bo!

[MathAndCode]

Here are some relatively small bakery predecessors.

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x = 6, y = 6, rule = B3/S23
4bo$4bo$bobobo$4bo$2bobo$2o!

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x = 4, y = 6, rule = B3/S23
2b2o$bo$2o$o$2o$ob2o!

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x = 6, y = 5, rule = B3/S23
2b2o$2bo$bob2o$o2bobo$4b2o!

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x = 4, y = 4, rule = B3/S23
o2bo$o$2obo$2b2o!

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x = 6, y = 5, rule = DoubleB3S23
2.AC2B$2.CBC$ACA.C$2A2.C$A2.A.B!

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x = 5, y = 5, rule = B3/S23
2bo$b3o$o3bo$4bo$3bo!