Who knew that Ed Sanford and John Townsend helped write songs for other artists like Michael McDonald with “I Keep Forgetting”? They still had one hell of a great song on their own with “Smoke of a Distant Fire.” The rest of that album is just as good, in my opinion.
Missed the edit: Take a look at this live performance from Burt Sugarman’s “Midnight Special.” I don’t understand how they weren’t more successful.
The Knack: M-M-M-My Sharona
The Romantics: That’s What I Like About You.
I was going to mention this. Great song … oddly singable to Beautiful Day by U2.
Safety Dance by Men Without Hats.
Did they even write any other songs? 
I checked up on Wiki, says it was a pretty big hit but in the UK (which usually transfers over to Germany) but only a minor hit in the U.S.
Marcy Playground - Sex and Candy
You have to be careful with these. This past summer Survivor played at a local festival, and many jokes were made on their behalf. “You know, Survivor! Eye of the Tiger! And…ummm…you know, Eye of the Tiger!” But once the show started, there were quite a few cases of “I remember asking Stacy McPretty to dance to this song in 6th grade!” and “Oh my god, we used to rock out to this 3-wheelin’ in the dunes!” It was an odd trip down the sidestreet of Memory Lane where the songs have no names.
Pop Goes the World 
In which they actually sampled “Safety Dance!”
Not American, but I’ve always really, really liked ‘Velvet’. It’s fun to sing.
Nena had a US hit with “99 Luftballoons” (the German language version doing better than the English version) and was pretty much never heard from again in the US. She’s still pretty prominent in Europe, though.
You are so clearly not Canadian.
Up here, anything from Gordon (especially Enid, Brian Wilson, or Grade Nine), their version of Lovers in a Dangerous Time*, My Old Apartment, Another Postcard**, and probably Alcohol or Who Needs Sleep would be easily recognized.
- Speaking of, aside from a version of I Saw Three Ships which was played to death at Christmas on the radio station played at my last job, I couldn’t name another Bruce Cockburn song if you payed me.
** My personal favourite. hums
He’s also Anthony Stewart Head’s brother.
Good Girls Don’t (But I Do)* was a lot better than My Sharona. So those are the two the OP called for.
My entry will be Procol Harum A Whiter Shade of Pale, followed by A Salty Dog
Okay, those two statements are causing me some cognitive dissonance. Really? You can’t name any other Bruce Cockburn songs? I’d better you’d know them if you heard them. Okay, well wait, maybe depending on how old you are. Certainly if you were the right age from the 80s, I’d think that you’d remember If I Had a Rocket Launcher, and Waiting for a Miracle.
The Refreshments had a song called Banditos back in the 90s, it was played incessantly on the radio down here in the southwest. Those of you in the PHX area would recognize the song, also that the band broke up and evolved into Roger Clyne and the Peacemakers. I guess they’re pretty big in that area since I see on their website that they are playing at the opening of the Tempe light rail station 
.
I doubt whether anybody my age (47) could name a single member of Focus, or a single Focus song besides “Hocus Pocus.”
But that one song was a classic, even if the guys in the band were serious prog-rockers who regarded that track as a goof.
I was born in '77, so (googling) I was 7 when Stealing Fire was released, and 10 when Waiting for a Miracle came out…somewhat younger than ‘the right age’, I suspect.
I only know about Lovers in a Dangerous Time because of BNL’s cover, for that matter.
I came in here just to mention the band “Toad the Wet Sprocket.” Most people, I believe, wouldn’t recognize the band name, bug just *might *recognize their few hits. These include “Walk on the Ocean” and “All I Want.” I totally freaked out my older brother by including a TTWS CD when he asked for some music to rip for his children who were just “discovering” music.