``Random spikes'' belong to the common language used by engineers, physicists and biologists to describe events associated with time records and locations in space. Data and signals consisting of, or structured by, sequences of events are omnipresent in communications, neurobiology, computer science and signal processing. We present a systematic study of the properties of random spikes and related complex signals, the latter being the result of various operations (filtering, jittering, delaying, thinning, clustering, sampling, and modulating) on a basic event stream. More specifically, we focus on second order properties, which are conveniently represented by the spectrum. The first contribution is theoretical: a modular approach for the construction of complex signals and formulas for the computation of the spectrum that preserve such modularity (each additional feature added to a basic model appears as an explicit separate contribution in the corresponding basic spectrum). Applications of the theoretical results are then presented: spectral formulas for traffic analysis, randomly sampled signals, and pulse based communications, including UWB ones.

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