1. He logged on to check his e-mail.

2. He logged in to check his e-mail.

3. He logged on to check his e-mails.
4. He logged in to check his e-mails.
---------------------------------------
Which ones are grammatical? Can we use all of them?

1 and 2 are fine, but normally the word ‘email’ is treated like a collective noun — no plural form needed.