Susceptibility measurements of patterned thin films at sub-K temperatures were carried out using a scanning SQUID microscope that can resolve signals corresponding to a few hundred Bohr magnetons. Several metallic and insulating thin films, even oxide-free Au films, show a paramagnetic response with a temperature dependence that indicates unpaired spins as the origin. The observed response exhibits a measurable out-of-phase component, which implies that these spins will create 1/f-like magnetic noise. The measured spin density is consistent with recent explanations of low frequency flux noise in SQUIDs and superconducting qubits in terms of spin fluctuations, and suggests that such unexpected spins may be even more ubiquitous than already indicated by earlier measurements. Our measurements set several constraints on the nature of these spins.