Hi,
* use a high speed OPAMP
* high frequency diode (low capacitance)
* limit OPAMP U2 output voltage during "clip" to avoid overdrive of OPAMP input stage. Read about OPAMP "phase reversal", "latch up" and "input overdrive recovery"
In your circuit, the OPAMP outut voltage usually is 0.55V below V+ inut voltage. But when the limiter is active the OPAMP output voltage (marked wit "2") immediately rises to positive supply rail. Maybe coupling from output to V- input causes the spikes.
The spikes could be because of
* traces capacitive coupling
* diode capacitance
* diode reverse recovery (time, charge)
Consider a low pass filter at the output
Also take care that the signal you see is really the signal of the limiter, and not effects of the scope´s jig.
Good luck
Klaus