There does not appear to be any charts or guidelines to this effect, and I cannot find a clear pattern by looking through the Monster Manual.
Tiny Creatures most often do a flat numeric value in damage, most often: 1. However, there are also Tiny creatures that deal d4 (Neogi Hatchling), d6 (boggle), or both (Velociraptor).
Small creatures range from d4s to d8s
Medium creatures range from d4s to d12s
Large creatures range from d4s to d12s
Huge creatures range from d6s to d12s
Gargantuan creatures range from d6s to d12s
Heck, the Tarrasque alone has attacks that deal d6s, d8s, d10s, and d12s damage
It's possible I have missed a few monsters that may expand these ranges even further...but perhaps I can shed some light on why this may be.
Note that this is mathematically based, not rooted in official word from WotC.
If you want a creature to do slightly more damage, you give it lots of smaller dice, rather than a couple of larger dice. This is because the more dice you are throwing, the higher your 'minimum roll' is without changing your maximum roll.
For example, let's look at the Wolf and the Giant Lizard. Both are CR 1/4 creatures, both have a +4 to hit and +2 to damage. The wolf rolls 2d4 for damage, the Lizard rolls 1d8.
Thus, the Wolf's minimum damage output is 4, while the Lizard's is 3. This curves their average damage per round upward, while not actually raising the maximum damage they can drop.
Against, say...someone wearing Chainmail, a Wolf's DPR is 3.40, while the Lizard's is only 2.90. However, both are still only able to max out at 10 damage. This means that over a prolonged fight, a Wolf is more deadly than a Giant Lizard. But its peak damage output is identical
On the other hand, if you made the wolf more deadly than Lizard by giving it a d10...it ends up with marginally higher average damage (3.65 vs AC 16), but now it can 'peak' at 12 damage instead of 10. Which, at first level, could be the difference between the party meat shield going "Holy crap, ow!" and "I'm rolling Death Saves."
For the purposes of the Monster Manual and the 'average damage' that it lists. It counts 2d4 as dealing the same 'average' damage as 1d10...but 1d10 is 'swingier' when actually rolled. It swings from 1 to 10, instead of 2 to 8.
So that brings us to where we are. A creature's average damage is a contributing factor to their CR. And playing around with dice size and number of dice lets you tweak their damage output to your liking...controlling both average damage output, as well as potential damage peaks.