Never seen this one before. It's neither Rampage, nor Movie Monster. I'm stumped :-/
[EDIT] Actually, I think der_baer_fm is right. I looks like Godzilla, although there is only one screenshot of it on Lemon64, and it's not that one. But it fits with the GUI with the left aligned square playing field, and the right aligned rectangular information window using the standard C64 font.

Actually its:
Crush, Crumble and Chomp!
I had it on tape i think, and my cousin wrote godzilla on it
Thanks Internet its:
http://www.mobygames.com/game/c64/crush-crumble-and-chomp/screenshots
heavy edited because my memory plays tricks on me

I'll give you one hint before I go to bed. The amount of damage you can inflict on the city (if you survive long enough) compares to the most expensive disaster movie epyx.
ha as I was writing we have a winner. Well close ENOUGH to a winner, typos don't count. It's technically correct name is Crush, Crumble & Chomp but I think yours is close enough.
Your go.
Also: If the xbox community games ever come to Australia, I'm going to get the XNA game studio and make a knockoff of CCC I think, that game was freakin awesome

Whats this Game?

great giana sisters, easy!

That looks like the Great Giana Sisters!
I don't think I've seen a more obvious clone of another game in my life.
[Edit: you can call me the one-minute-late man!]

That was damned easy! AliasAlpha, time for one more before bedtime?

A pirated copy I'm proud to say I had. How dare nintendo try and stifle the creative freedom to shamelessly, obviously and comprehensively steal the creative efforts of someone else?
okay, I'll be a bit generous and give 2 screenshots for this one. Mostly so you can check out the rack on the brunette
Should be relatively easy as it was a moderately big game at the time. Oh and a strange bug means you'll see the screenshots twice, oh well makes it even easier...

Rocket Ranger, for the Amiga :-D
[EDIT] Well, AliasAlpha has apparently gone to bed. In the interest of keeping the thread going, I'll announce myself the winner of this round. Here are a few links to back up my claim:
Rocket Ranger at Mobygames
Rocket Ranger at LemonAmiga
Here's an easy one to get the game going again.

Ooh look at him, proper little megalomaniac, declaring himself this and that...
Well there's no denying that you're RIGHT so I guess it's justified. Gotta love the old cinemaware games, they did some great stuff, also had a bit of a boobie fixation (bet his skull wasn't the only thing that was throbbing), also a bit shameless in their cross-promotion...
I'll stop guessing for a while now, let other people have a go

Hehe, well, since nobody jumped on the band wagon while you were away, I could have saved myself the trouble.
This one, however, is also very easy. It's It Came From The Desert. Bit of a Cinemaware fan, aren't you?
[EDIT] And don't stop guessing. If anyone wants in the game, they can jump in any time they like.

Well yeah they were pretty great. But its not like I guessed the game you were asking about, I just posted that as a cheap excuse to show off shameless cross promotion and exaggerated nursey cleavage.

Run that past me again, would you? You didn't guess the game?
[EDIT] Not trying to be an asshat here, I'm genuinely baffled, as it seemed to me that you did.

Populous?
I never managed to play It Came from the Desert. I had no Amiga and i didn't seem to find the pc version for sale. :(