The hybridisation and therefore shape of a radical can be anywhere between sp³ and sp², although the latter is the more common (with the unpaired electron in the p orbital) - there is some preference for it, shown by the difficulty of forming radicals in bridged species that cannot reach sp² hybridisation.
Radicals are not particularly stable in general, as noted before, so their stability is only a relative term used to compare radicals with radicals. They can be stabilised by;

Steric Relief: If the radical to be formed is planar, then for centres with bulky substituents there will be steric relief obtained in forming the radical - as with carbocations (see the nucleophilic substitution section on these), though to a lesser extent.

Conjugation/Resonance: Again, as with both carbocations and carbanions, a radical that can delocalise across π systems will be stabilised. It should be remembered that resonance may allow the radical to react at a different centre to the one at which it was formed.

Steric Protection: This is not technically stability in thermodynamic terms, but more in kinetic terms - if there are sterically bulky groups around the radical centre, it will be difficult for anything to reach the radical and react with it - conveying an apparent stability. In fact, these radicals are quite likely to be the ones trapped and observed, because their steric protection prevents them from reacting at all.

Weakness of the original bond: For example whilst oxygen is not particularly stable in a radical state, relative to the rather weak O-O bond oxygen radicals are not disfavoured - similarly for other weak bonds mentioned on the previous page.