Take a look at ebay item 140271683808 - I bought a pack of these tungsten carbide dental burrs a while back and found some of the shapes very useful for texturing - but the selection also includes cylindrical shaped ones that should work for piercing (ie side-cutting) although I've not had reason to try these as yet.
They are very cheap, nice and small for fine detail, and fit a dremel collet.
They are actually cutters, rather than burrs and the cylindrical ones look like tiny milling cutters. I've not found they burn the wood (although it's worth experimenting with the speed control on your dremel-type tool, as I've found there seems to be an optimum speed for cleanest cut depending on the wood you are using) and the ball & dovetail shaped ones I've tried for texturing make really nice clean cuts rather than tearing the wood.
The only downside is that being tungsten carbide they will chip & break very easily if knocked against something hard (like part of a lathe!) so you need to be somewhat careful when putting the dremel down with one installed in the collet.
Kym