Wavelets seem to be very useful for image processing. Assuming that I'm only ever going to study signals of time, i.e. 1D signals, should I still take a course in wavelet analysis? Are they applicable to 1D signals?
They absolutely are applicable. They are used for signal compression, denoising, feature extraction, filter-bank signal processing and many more things. In fact, a course of wavelet analysis will (should) stress one-dimensional analysis, which 2D is an extension of.