[FrankenMauser]

A dedicated rimfire can is definitely the way to go.
Rimfire ammo, with very few exceptions, is absolutely filthy and any suppressor that sees rimfire use must be cleaned frequently.

Centerfire rifle cans, however, don't really need to be cleaned (or even capable of being cleaned by the user) unless you shoot cast bullets.

There are a large number of .223/5.56 cans on the market, ranging from garbage to excellent. Unfortunately, I don't know that market segment well enough to make a good recommendation. (For centerfire, mine are .30 cal or larger.)

For rimfire, I like the SilencerCo Sparrow .22 and Tactical Solutions Ascent .22. They are both good suppressors, affordable, well made, light weight, small diameter*, and easier to clean than their peers. *(I can still use the sights on my Buckmarks, with the Ascent; where most rimfire cans would require more height - like a red dot.)
Even those two very similar suppressors aren't exactly alike, though. The Sparrow can handle some hotter cartridges, like .22 Hornet and 5.7x28mm; whereas the Ascent is rimfire-only (but lighter).

Look around, read reviews, see what interests you.

One small bit of advice that I will offer for rimfire, is to ignore the Bowers Bitty. Though an affordable, adorably small, light rimfire suppressor, it is not really suitable for use on a pistol. It was designed with rifle-length barrels in mind, and that is where it belongs. It doesn't offer sufficient sound attenuation on pistols.

And a note, on the market, overall: Bowers doesn't get talked about much, but their suppressors are well made, well liked, and their customer service is amazing.