How to detect a spiky signal embedded in a periodic signal Say I've got a signal that I know is composed of periodic components. However, I'm worried that there are "spiky" events that are embedded in this signal, and potentially occurring periodically. I can't just look at the power spectrum of the signal, because the spiky event will show up across many frequencies in the power spectrum.

What kinds of methods are out there to tease apart components of a signal that can reasonably be decomposed into sine waves, from those that cannot (like a spike)?
You can use linear prediction: look at the residual of an AR model. If the signal is a sum of sinusoidal components - and provided the order of the model is appropriately chosen - the prediction error will be small. Unexpected spikes will directly show up in the error signal.