Stability is a relative concept in chemistry. In the lab, we tend to use the term a bit loosely, and we mix up with “reactivity”. But those are two axes of an x-y graph, and there are chemicals in all four quadrants. Stable and non-reactive? Sure, for whatever value of “non-reactive” you choose. Stable and reactive: how about acid chlorides? You can keep many of them happily for years away from water, amines, etc., but open the flask and they’ll be ready to party. Unstable and non-reactive? An odd category, but I’d say that something like a polyazide or polynitro compound would fit. It doesn’t do much with other chemicals; it just falls apart on its own, and how. And unstable and reactive? Oh, yeah, we have those, all right.
In the lab, there’s a large middle ground of things that sort of gradually deteriorate on you, but not so quickly as to be a nuisance. Solutions that used to be clear pick up a yellowish cast, crystals get cloudy. This is the sort of stability that people are used to seeing with newsprint paper and household chemicals like bleach – they’re good for a while, but you can’t expect them to hang around forever. In research, you deal with this by either buying new stuff (the industrial way!) or re-purifying the old bottle by distilling or recrystallizing it (the academic way, by necessity).
After these compounds, though, you come to the ones that can give you trouble. There are a lot of compounds that are only stable on a time scale of days, hours, or minutes, and you’ve got to keep an eye on these guys. Often the rate of decomposition is very dependent on how pure the stuff was at the beginning. Trace amounts of water, oxygen, or other such rare substances can start one of these down the slope.
The dangerous ones are the compounds whose decay begets their own decay. These will run away on you, and if there’s enough compound in the flask where heat transfer is a problem, the process can turn violent. At this point, we’re shading over from “troublesome decomposition” to “explosive hazard”. Things like this are best kept as cold as possible, and in dilute solution. Concentrating them or warming them is a deliberately provocative act for which payment will be due.
Even without explosions, this sort of thing can be alarming. I’ve heard of intermediates that were so lively that initial clearish substances in a round bottom flask turned brown and began to fume as the person walked down the hall holding the stuff. Generally, that only happens once, the first time you make one of these beasts. After that, you take appropriate precautions (like having the next reaction step set up right next to this one, ready to go). Or, of course, you just decide that you can live without that one, and never make the darn stuff again.