I've simulated a nice circuit for my purposes and find I can't breadboard it. I've checked continuity on all connections before powering, but I've failed to get it running.
There is no decoupling capacitor from the +ve supply to the 0v (Ground) line. Try a 10nF.
At a quick and dirty calculation, the oscillator frequency appears to be of the order of 2MHz.
Some breadboards can be a bit funny at high frequencies, although at 2MHz I think that you may "get away with it".
What frequency does the simulation say?
The simulation is saying 20kV on the secondary of the transformer, that is a bit much for any plug-in breadboard with 0.1" contact strips.
For 0.1" plug-in breadboards, 50vV would be a sensible maximum voltage.
I suggest that you build your circuit in an "ugly" or "Manhatten" style on a piece of circuit board with a continuous copper ground plane.