There are two considerations when removing paint from a miniature: the type of mini (metal vs. plastic), and the solvent to be used.
Metal miniatures are much more resilient, as as such can withstand harsher chemicals than plastic minis. The plastic used for miniatures is usually fairly soft, so you have to be very careful about letting it sit in a chemical bath for any length of time!
I have found that undiluted Simple Green (a household cleaner, which can be found at department stores) does a good job of dissolving the old paint. I've left metal miniatures to soak in Simple Green for a day or two (even as long as a week), and it usually causes the paint to lift off of the metal. I haven't tried it on plastic miniatures, and though I've heard it's safe, I wouldn't leave them unattended for longer than a day (if that). Simple Green has a "pine-ish" scent, which is rather potent (given that it is undiluted). I try to leave it outside when I can (a back porch is ideal).
Previously, I have used nail polish remover on metal miniatures. Do NOT use this on plastic miniatures, as they will most likely melt! It works, but my results were not as effective as Simple Green. The smell is even more noxious than Simple Green, as well.
I've read warnings about using vinegar on metal miniatures, as the acetic acid can mar & pit the metal. If you decide to try vinegar, I would not use it as a chemical bath, but only as a spot cleaner.
Another solution I've heard is dishwashing soap. I'm giving this a trial run now, and will post the results.
If you wish to accelerate the paint removal process, an old toothbrush can be used to scrub the miniatures. Coupled with the chemical action of the solvent, the physical scrubbing can really speed up the process. I've read that the stiff, narrow brushes designed for cleaning braces are even more effective, and I look forward to trying one out.
Note that paint removal may often take multiple "sessions", as the solvent works on the outermost layers. Again, scrubbing the minis can accelerate the process, as it helps remove the outer layers and allow the solvent to reach the next paint layer more quickly. One of the worst cases I've encountered is where miniatures have two layers of primer, which is harder to get up than paint. In such a case, I have used a second chemical bath; if you do so, be sure to dispose of the used solvent and use a fresh batch to make sure the process will do any good!
UPDATE: It seems the dishwashing soap did absolutely nothing to help remove the primer on my test miniatures. It might do better against actual paint, but as far as removing primer, I'll be trying a second bath of Simple Green.