iOS: Unless you're using a tripod, shooting videos on your iPhone or iPad will produce some camera shake. Emulsio is a neat app that smartly stabilises the videos you shoot to make for a shake-free viewing experience.
The app lets you enable or disable stabilisation for each axis; compensate for rotational camera motion; remove wobbling artifacts; and use a "stabilisation strength" slider bar to decide how much it should zoom in to reduce shakes around the edges of the screen. The really cool part is that you can split the screen into two to compare what the video looks like before and after applying the effect.
In the free version, the app will only work with 30-second videos and watermarks them when exporting to your Photos library. You'll want to unlock the app to get unlimited time and no watermarks, which is an in-app purchase of $1.99.
Emulsio uses videos from your Photos library; it doesn't record the videos itself. Video Camera+ is still the best video shooting app for iOS, although you should combine it with some tricks to shoot better video on smartphones. And then use Emulsio to stabilise your footage!