3.2. Growing-mode perturbations
This is the key prediction of inflation as a theory of the origin of structure; inflation generically predicts oscillations in the temperature and polarization angular power spectra. If oscillations are not seen, then inflation cannot be the sole origin of structure.
The reason this prediction is so generic is because inflation creates the perturbations during the early history of the Universe, and they then evolve passively until they enter the horizon in the recent past. The perturbations obey second-order differential equations which possess growing and decaying mode solutions, and by the time the perturbations enter the horizon the growing mode has become completely dominant. That means the solution depends only on one parameter, the amplitude of the growing mode; in particular, the derivative of the perturbation is a known function of the amplitude. The solution inside the horizon is oscillatory before decoupling, and this fixes the temporal phase of the perturbations; all perturbations of a given wavenumber oscillate together and in particular at any given time there are scales on which the perturbation vanishes. Projected onto the microwave sky, this leads to the familiar peak structure seen in predicted anisotropy spectra, though if one wants to be pedantic the troughs are if anything more significant than the peaks.
The importance of the peak structure in distinguishing inflation from rivals such as defect theories was stressed by Albrecht and collaborators  and by Hu & White . The prediction is a powerful one; in particular it still holds if the inflationary perturbations are partly or completely isocurvature, and if they are nongaussian.
I stress that while inflation inevitably leads to the oscillations, I am not saying that inflation is the only way to obtain oscillations. For example there are known active source models which give an oscillatory structure , though the favourite cosmic string model is believed not to. If observed, oscillations would support inflation but cannot prove it. However I might mention in passing that it is quite easy to prove  that the existence of adiabatic perturbations on scales much larger than the Hubble radius would imply one of three possibilities; the perturbations were there as initial conditions, causality/Lorentz invariance is violated, or a period of inflation occurred in the past.
Against designer inflation
At this point it is worth saying something against `designer' models of inflation which aim to match observations through the insertion of features in the spectra, by putting features in the inflationary potential. This idea first arose in considering the matter power spectrum , which is a featureless curve and so quite amenable to the insertion of peaks and troughs. However the idea is much more problematic in the context of the microwave anisotropy spectra, which themselves contain sharp oscillatory features. One might contemplate inserting oscillations into the initial power spectrum in such a way as to `cancel out' the oscillatory structure, but there are however three levels of argument against this: