Can you focus the scope trace?
Also zoom on the spikes, they look ~sort of~ consistent, not that there should be any reason for such a thing to be produced randomly from a linear supply regardless. A DSO (or single sweep + long exposure in dimly lit room) would be helpful here.
Possible culprits: electrical breakdown (capacitor? diode/transistor avalanche?), coupling (AC line noise? nearby transients?), frayed wires (intermittent shorting or breakdown), crummy resistors..?!
Aside from aging capacitors (electrolytics are always prime suspect), almost anything else is a stretch.
Tim