What is the state-of-the-art of using signal-processing for "cleaning-up" videos?
Entertainment industry greatly exaggerates what video enhancement can do.

Spatial and temporal noise reduction

Noise reduction by Neat Video and MSU Denoiser Filter, which are somewhere at (according to a comparison from 2007) or near state-of-the-art for hard noise, takes advantage of that adjacent frames usually are of the same scene but with independent noise. So noise can be filtered both spatially and temporally. But that kind of processing is in my experience just cosmetic. It does not really reveal things that a human would not already see through the noise, it just makes the video look nicer. Both filters have an option to adjust settings automatically, but because noise reduction seems always a compromise between noise reduction and introduction of artifacts, the final "strength" of noise reduction should be adjusted to taste manually.

Motion estimation and superresolution

MSU Super-Resolution Filter reconstructs a new video at 2x horizontal and vertical resolution. It follows a similar paradigm as the above two filters, but also estimates motion between frames. The original pixelization (and other artifacts that do not follow the motion such as blocking and sensor Moiré) can be reduced by compensating for motion and then filtering temporally. This improves resolution and makes detailed things like pixel-deprived text and faces more clear. Resolution can be increased only if the pixelization is not properly anti-aliased. Artificial pixelization of faces, if not done carefully, could be reduced by the same or similar processing. This is a kind of enhancement that the human visual system is not accustomed to do natively, and can actually reveal hidden useful information.