Which comes first: cat or dog?

Which comes first in a sentence?

I know some word pairs such as bacon and eggs, where bacon always comes first. E.g:

Make me bacon and eggs for dinner, honey.

Cats and dogs are the two most common pets I know of. What is the preferred order?

1. Dogs are superior.

   Don't put dogs and cats together, they might fight!

2. Cats are superior.

   Don't put cats and dogs together, they might fight!

Given the phrase it's raining cats and dogs, you might be led to believe that this is the usual ordering. However, COCA gives the following frequencies:

cats and dogs    272
dogs and cats    359

So it seems there isn't really a preferred ordering.

While "cats and dogs" is common, especially in popular idioms noted by others ("raining cats and dogs", "fighting like cats and dogs"), it would not surprise me in the least if the "dogs and cats" order received a significant boost from people quoting Bill Murray's character, Dr. Peter Venkman, in the 1987 movie Ghostbusters:

Human sacrifice, dogs and cats living together... mass hysteria!

Very handy when you want to humorously indicate the potential for great catastrophe, or perhaps more commonly, sardonically when you think the dangers are being exaggerated.

Regarding any pair of nouns in general, if you're married, just ask yourself how your parents refer to you and your partner. Compare that to how your partner's parents refer to the two of you.

In a bygone age it might have been standard practice for both sets of parents to name the man first, but today each parent names their own child first. And usually no-one even notices both naming systems being used concurrently at family gatherings and the like.

The basic point is people usually name the more important of any such pair first.

Regarding cats and dogs, apart from the common idiom that it might be raining them in that order, there's no significant preference either way for other contexts. As the chart shows, 'twas ever thus. Some people are cat people, some prefer dogs. They tend to name their preferred pet type first.