DIY colpitts oscillator not oscillating
Breadboard is likely to have higher parasitic capacitance between tracks than either of your filter capacitors (10 pF and 5 pF) - this means trouble
The 100 uH inductor is quite possibly self-resonating below the frequency you are aiming for and this means you won't get sustained oscillations.

For both of the above try lowering the inductor value (L1) by ten (or more) and increasing C2 and C6 by ten (or more).

L3 at 1 uH should be significantly higher than L1 for proper operation.

Choose inductors that have a self resonant frequency (SRF) above the desired oscillation frequency. If the data sheet for an inductor doesn't mention SRF, don't use that inductor.
One last point - the values you have in your schematic are for running at around 8.7 MHz - is this what you intended? I point this out because an FM receiver runs at 88 to 108 MHz.