The US-released games that don't work on TMSS models are Budokan, Ishido, Onslaught, Populous, and Zany Golf. The latter four can be made to work by using a Game Genie as a passthrough cart, but Budokan won't run on a TMSS Genesis no matter what. However, there's a European version that's been fixed.
In addition, there are a handful of Japanese games that don't comply with TMSS -- mostly launch versions of well-known titles, plus the strategy game Super Daisenryaku -- and also a bunch of Asian and Russian pirate games, only a few of which are worthwhile and language-accessible.
The usual wisdom is that the best Genesis to get is a Model 1 with "High Definition Graphics" and no TMSS, but as mentioned, it's really the VA7 motherboard that has sound problems. This guide goes into more detail, though the images are broken, and I also seem to remember that there was a problem with some (most?) VA2 Model 1 consoles only putting out one channel over the AV connector (i.e. putting out the R channel as mono, instead of summing L+R to mono).
For all the hand-wringing over this stuff, I personally use a Model 2 or Nomad 99% of the time. The Model 1 has great sound, but you can't get stereo out of the back of an unmodded Model 1, and I find that the headphone jack is prone to very loud popping on power on/off. The Nomad has excellent sound quality, and while the Model 2 sound quality is definitely subpar, if I use a 32X with my Model 2 the image quality is terrific (since the Model 2 sends RGB to the 32X, which has a superior composite encoder).
There's also a mod for the Model 2 to supposedly make it sound great, but I can't vouch for it, and some report that the modded sound is overly bright and trebly (which wouldn't surprise me). Also, late revisions of the Model 2 allegedly have much better sound quality, but again I can't vouch for that.
BTW OP, if you need any Genesis games, let me know -- I have half-a-dozen CIB doubles of quality games, and they're surely not doing me any good just gathering dust.