grammar about the verb "to suggest"

Seraphin

I seem to recall such a rule in the English grammar (using the example to illustrate it)

example: the evidence suggests that he (should) be the one who does it.

but can we say

a) "the evidence suggests that he MAY be the one who does it" ?

b) "the evidence suggests that he CAN be the one who does it" ?

c) "the evidence suggests that he IS the one who does it" ? 

It seems to me that in all of these examples "does" should be replaced with "did". After all, evidence is only gathered after some event. With that correction, out of the four suggested words only "may" and "is" would strike me as correct.

Ray is right about the tenses. A more normal context would be:

He suggests that she (should) be the one who does it.

should be or be 

are subjunctives implying an obligation ("ought to") and cannot be replaced by the verbs you suggested.

This is the so-called putative should.  Do some searches for it (top right Search box on this page), there are many threads on it. Read some of them. 

 Bokeh's sentences

 Someone keeps peeing on the back fence. The evidence suggests that he may be the one who does it.

are OK, but I don't think they correspond to the original one

  the evidence suggests that he (should) be the one who does it

that I think should be anyway modified and clarified.