It all depends on what you mean by stability...

Short Term Stability, over the next couple of years?
- Stable relationship with all major partner(s)
- No financially endangered (ie BK) major partners.
- Plenty of cash on hand...lenders are NOT putting out right now...if you get in trouble, you will have to bail yourself out (unlike 2001-2007).

If you're looking for Long Term Stability (ie career destination), then you have a different set of criteria:
- Large: In case anyone hadn't noticed, size matters and large airlines (both regional and major) are the future. There are a few niche markets for national airlines (ie allegiant), but there is no benefit at all to being a small regional.
- Multiple Major Partners...diversity is critical if something bad happens.
- The Expertise and Experience to acquire new contracts, and implement them. This is another place where small companies tend to suck...mom, pop, and bubba have a hard time running the operation they already have, to say nothing of starting up new operations...
- Good, conservative management track record.
- Larger RJ's...the more, the better.

Actually if you want Long Term Stability, better get a job at a major.