Before we get to those, lets get some of the dafter suggestions out of the way.

The phrase seems isn’t related to the well-known antipathy between dogs and cats, which is made word in the phrase "fight like cat and dog". Aside from the presence of cat and dog in the phrase, there’s nothing at all to connect their fighting with raining.

Nor is the phrase in any sense literal, i.e. recording the fact that cats and dogs fell from the sky. Numbers of small creatures, of the size of frogs or fish, do occasionally get carried skywards in freak weather. That must happen to individual dogs or cats from time to time too, but there’s no record of groups of them being scooped up in that way. Not that we’d need meteorological record for that – it’s plainly implausible.

In fact "raining cats and dogs" makes sense figuratively and the explanations below that attempt to link the phrase to felines, canines and weather seem rather feeble.

Here goes though – take your pick:

It comes from mythology. Witches, who often took the form of their familiars – cats, are supposed to have ridden the wind. Dogs and wolves were attendants to Odin, the god of storms and sailors associated them with rain. Well, some evidence would be nice. There doesn’t appear to be any to support this notion.

Cats and dogs were supposed to be washed from roofs during heavy weather. This is a widely repeated tale. It got a lease of life with the message "Life in the 1500s", which began circulating on the Internet in 1999. Here’s the relevant part of that:

I’ll describe their houses a little. You’ve heard of thatch roofs, well that’s all they were. Thick straw, piled high, with no wood underneath. They were the only place for the little animals to get warm. So all the pets; dogs, cats and other small animals, mice, rats, bugs, all lived in the roof. When it rained it became slippery so sometimes the animals would slip and fall off the roof. Thus the saying, "it’s raining cats and dogs."

This is nonsense of course. It hardly needs debunking, but, lest there be any doubt…

Dogs lived in thatched roofs? No, of course they didn’t. Even accepting that mad idea, for them to have slipped off when it rained they would have needed to be on the outside – hardly the place an animal would head for to shelter from bad weather.

The phrase is supposed to have originated in England in the 17th century when city streets were filthy and heavy rain would occasionally carry along dead animals.

The idea that seeing dead cats and dogs floating by in storms would cause people to coin this phrase is just about believable. People may not have actually thought the animals had come from the sky, but might have made up the phrase to suit the occasion.

Another suggestion is that it comes from a version of the French word, catadoupe, meaning waterfall.

Well, again. No evidence. If the phrase were ‘raining cats’ or if there also existed a French word, dogadoupe we might be going somewhere with this one. As there isn’t lets pass this by.
Returning to facts rather than idle speculation, we do know that the phrase was in use in a modified form in 1653, when Richard Brome’s The City Wit, has the line:

"It shall raine … Dogs and Polecats".

Polecats aren’t cats as such but the jump between them in linguistic rather than veterinary terms isn’t large.

In a form more like the current version it appears in Jonathan Swift’s ‘A Complete Collection of Polite and Ingenious Conversation’ in 1738:

"I know Sir John will go, though he was sure it would rain cats and dogs".

More likely than any of the versions given above is that this is just a nice descriptive turn of phrase, which doesn’t relate to any particular event or practise.

There’s a similar phrase originating from the north of England – "it’s raining stair-rods". No one has gone to the effort of speculating that this is from mythic reports of stairs being carried into the air in storms and falling on gullible peasants. Its just a rather good vivid phrase giving a graphic impression of heavy rain.

Another similar phrase is "it’s raining like pitchforks", the first known reference of which is D. Humphreys’ Yankey in England, 1815:

"I’ll be even with you, if it rains pitchforks – tines downwards."