Quoting the Wikipedia article on "Shaky Camera":
Handheld camera, shaky cam, queasy cam, queasicam, run-and-gun or free camera is a cinematographic technique where stable-image techniques are purposely dispensed with.
The camera is held in the hand, or given the appearance of being hand-held, and in many cases shots are limited to what one photographer could have accomplished with one camera.
Shaky cam is often employed to give a film sequence an ad hoc, electronic news-gathering, or documentary film feel. It suggests unprepared, unrehearsed filming of reality, and can provide a sense of dynamics, immersion, instability or nervousness. The technique can be used to give a pseudo-documentary or cinéma vérité appearance to a film.
You can accomplish this in effectively the same way as camera shake in response to a catastrophic event, just at a lower amplitude. You track a nominal camera position and orientation, then displace the camera's actual position and orientation used for rendering using a pseudo-random function.
Generally this looks best if you use a continuous function, like a Perlin noise or even sine waves to emulate periodic vibration. This keeps the camera from jumping around too distractingly, and lets you play with a range of frequencies from slow sway at a medium amplitude to a rapid tremble at low amplitude. Mixing samples at multiple periods and amplitudes can give the motion a more complex, less mechanical-looking character.
You can also play with how you distribute the shake between translational & rotational offsets:
translational shake affects the foreground more strongly than the background, with up/down/left/right shake being the most noticeable, and shake forward/back usually looking more subtle.
rotational shake affects the whole image together, so it can reduce the appearance of the foreground vibrating but also has more risk of disorienting the viewer. Keep the amplitudes lower, especially for roll offsets, or you might make players seasick.