Just had to deal with something like this the other weekend. First thing I’d say is to stay away from chemical drain cleaners, if they don’t do the trick right off the bat then you may have a pipe full of caustic chemicals to worry about while you’re fixing your drain. Save them for the absolute last resort, trust me.
Do you have a double-basin sink? If so is the stink ONLY in the disposal side or is it also in the side where the regular drain is?
If it’s only in the disposal side then probably a buildup of stinky crap in the disposal. Dunno how to clean them out if ice cubes, lemon wedges and baking soda don’t do the trick, might need to dismount it and clean it out. You can try pouring boiling water down (be careful!) and see if that dissolves any “stuff”.
If you run lots of water down the disposal does it backup into the disposal basin? If so then there’s probably a clog in the little pipe connecting the disposal to the drain and it’ll need to be cleared out.
If the stink comes out of both basins or is really noticeable from the “regular” basin then it’s probably coming from the main kitchen sink drain.
If you run lots of water down the drain does it back up at all? If not then you don’t have a major clog. If it does then you need to clear out the drain.
Either way here’s what I’d try. Get yourself a bucket, some old towels, a pipe wrench or channelock pliers and if you don’t like this kind of work find a friend/spouse who does (beer makes a nice bribe).
Clear out all of the bottles and whatnot from under the sink and put the bucket under the J-shaped trap (that’s the bent piece of pipe that holds some standing water which prevents odors from coming up out of the drain and into your sink). If that trap is leaking (run water and watch it for a minute) then that’s the trouble - if the trap doesn’t hold water then nothing blocks the stinks. You’ll need to tighten the bolts or replace the trap or otherwise fix the leak as appropriate.
If everything looks good put the bucket under the trap to catch any water when you remove the trap. Carefully undo the bolts holding the trap in place (I’ve had to use a wrench before, othertimes I could do it barehanded) and remove it. Empty the water out and then check the trap to make sure it isn’t coated with crud. If it is clean it out (bent coathanger should make a good pipecleaner), flush it clean and reinstall, see if that takes care of the problem. If there’s no trap down there then you need to get one installed, pretty simple task for a plumber.
If that still doesn’t do it then there may be a buildup of nasty stuff further down the drain. This can be cleared out with a plumber’s snake which is a very cheap item down at the hardware store or you can have a plumber do it. I’m not sure how a bad smell from downstream of the trap could get back past a clean and properly-functioning trap but who knows what evil lurks in the hearts of the plumbing…
I’m not a plumber but that should get you started and give minimal chance of biblical flooding in your kitchen.
Worst that you have to do is put the trap back in place and call a plumber.