Instability is one thing, oscillation another. Then there are different types of oscillators - for example, a monostable multivibrator is a different beast compared to a harmonic oscillator. A harmonic oscillation a fine balance between stability and instability, where the phase space trajectory closes on itself. OTOH an unstable system exhibits a phase space signature that spirals out.

Assuming you are interested in harmonic oscillations, poles on the jω axis are only a (rough) necessary condition for oscillation. A harmonic oscillator is an inherently non-linear circuit i.e; even though the circuit elements appear linear, the amplitude of the oscillation is determined by a saturation of the negative resistance characteristic. That is, the oscillations grow in amplitude until any further growth leads to a drop in negative resistance and damps out the oscillations. Further, due to pulling, the oscillation amplitude ends up shifting the frequency as well.

We electrical engineers get too caught up in linear analysis because it's easy, but there is plenty to life beyond jω.  An excellent tutorial on oscillators can be found at:

http://www.uni-potsdam.de/u/phys_gprakt/html/projekte/emwellen/EM-Schwingungen_1...

M.G.Rajan
www.eecalc.com

It's what I call the AWGN approach - say a lot and you can be sure something will match