>>> Are they legit ?
Almost certainly not. It's easy to spoof the address shown in an email. (You need to go into the detailed headers to find out where it's really from). So the fact that an email appears to come from a 'gov.uk' address is meaningless.
Any genuine communication is still likely to arrive by post. Alternatively you might get an email asking you to log into your HMRC account (for example) in order to read a message there. (Any such email will address you by your full name. e.g. "Dear Henrietta Worthington", rather than as, say, '"Dear Customer").
>>> Has anyone else had them ?
Probably about half the adult population of the UK! Such emails are extremely common.
If you want a more definite answer, please let us know which government department the email purports to ome from, whether you're addressed by name and exactly what it says. (In the meantime, do NOT click on any link in the email!!!)