What physical or mental actions can be picked up by EEGs?
Many EEG responses are swamped in random brain activity, artifacts and background noise. A single movement typically doesn't evoke measurable activity, because its amplitude is so small with respect to background noise. 

I think the potentials you are looking for are event-related potentials or ERPs (Fig. 1). During ERP recording, basically a regular EEG is registered, but certain events (presentation of stimuli or actions by the subject) are time locked with the EEG recordings. By repeatedly measuring (e.g., 25 times) the time-locked response to the event, an average EEG can be obtained. Assuming remaining EEG activity, noise and artifacts are random, the averaging process will decrease noise, but will not affect the ERP. The averaging hence leaves you with a clean ERP response (Yordanova et al., 2003).


Fig. 1. ERPs. Left are stimulus-evoked ERPs (auditory, visual) and right are ERPs associated with the motor response by the subject (button-press upon stimulus presentation). Source: Yordanova et al., 2003.

In Fig. 1, the reaction times indicated correspond to the subject's button press, so the intention to press the button in fact precedes the actual motor response. In fact, the covert (imagined) ERP response can be nearly identical to the overt ERP (the motor action), see Fig. 2 (Kranczioch et al., 2009).


Fig. 2. Overt motor-evoked ERPs (top three panels) and equivalent covert (imagined) ERPs (lower three panels). The three panels cover different electrodes. Source: Kranczioch et al. (2009).

By characterizing the exact shape and other characteristics of the ERP, computer software can be deployed to recognize this ERP wave within a normal EEG. This is done in computer-brain interfaces. By filtering the ERG outside the frequency range associated with the ERP a lot of the cleaning can be done on the fly in near-real time.   

A list of various stimuli in ERPs can be found in Table 1 in (Goodman, 2010).

References
- Goodman, Atten Percept Psychophys (2010); 72(8): 10.3758
- Kranczioch et al., Human Brain Mapping (2009); 30(10): 3275–86
- Yordanova et al., Brain (2003); 127(2): 351 - 62