I have 4 convictions for driving offences. 2 are over ten years ago, 1 is 9 years ago and 1 is 4 years ago. Can I apply for a temporary residents visa?

What country?

I know Canada uses the exact term "temporary resident visa" for their visitor's visas so I'm going to assume you mean Canada and it would depend.

What do you mean "driving offences"? If you mean DUI then you will have problems. Canada considers DUI to be a much more serious offence than the US does and you may not be allowed a visitor's visa (or any temporary visa).

Again, more information is needed to give you specific advice.