Linguistically, the group of modals in English function as pairs: will / would; can/ could; should / must; may / might. Interestingly, this is also their order of frequency. So “will” is the most frequent, followed by its cousin “would”; “would” is more common than “can”, which is more common than “could”, and so on. In general, modals:
1) are a closed set: will / would; can / could; should / must; may / might (ranked by frequency)
2) do not co-occur in a sentence: *I don’t can play piano
3) do not take ‘s’ in the third person: *He cans play piano
4) are statements of speculation (‘modality’) not fact.
5) most commonly co-occur with the verb “be” and verbs meaning “think” (imagine, consider, guess, etc).
According to Biber et al (1999) other verbs commonly co-occuring with modals (>60%) are: admit, afford, appeal, assure, cope (with), guarantee, handle, imagine, resist, settle for, suffice, survive, tolerate.
You can imagine what that’s like.
I can’t cope with this.
The council cannot afford to maintain them.
This highlights two important features of modal pairs; the first one (will, can, may) expresses more certainty than its pair (would, could, might).
We will see an economic recovery in the next few weeks.
The second in the pair, on the other hand, expresses distance, whether spatial or temporal: We would have done the job if we had had the time. “Would” expresses past time, which “will” cannot. Would you please close the window? is also more polite than “will you…”
Modals can also be sorted into three functional groups: prediction / volition (will, would); possibility / permission (can, may) and, finally, degrees of obligation (should, must).
For more information, see Modal Verbs in the Grammar Resource.