We derive relations between standard order parameter correlations and the noise correlations in time of flight images, which are valid for systems with long-range order as well as low-dimensional systems with algebraic decay of correlations. Both bosonic and fermionic systems are considered. For one-dimensional Fermi systems we show that the noise correlations are equally sensitive to spin, charge, and pairing correlations and may be used to distinguish between fluctuations in the different channels. This is in contrast to linear response experiments, such as Bragg spectroscopy, which are only sensitive to fluctuations in the particle-hole channel (spin or charge). For bosonic systems we find a sharp peak in the noise correlation at opposite momenta that signals pairing correlations in the depletion cloud. In a condensate with true long-range order, this peak is a delta function and we can use Bogoliubov theory to study its temperature dependence. Interestingly we find that it is enhanced with temperature in the low-temperature limit. In one-dimensional condensates with only quasi-long-range (i.e., power-law) order the peak in the noise correlations also broadens to a power-law singularity.