I was born in 1970, so the 80's are my musical heritage.
Some songs, many not yet mentioned, that were my favorite song at one point in time or another:
Call Me - Blondie
We Got The Beat - GoGos
In The Air Tonight - Phil Collins (already mentioned I believe)
Running Up That Hill - Kate Bush
In Your Eyes - Peter Gabriel
All You Zombies - The Hooters
Drive - The Cars
Heartbeat City - The Cars (sorry - I just really love this song)
Cruel Summer - Bananarama
Cities in Dust - Siouxsie & the Banshees
Where is My Mind - Pixies
West End Girls - Pet Shop Boys
Time Stand Still - Rush (yeah, I know)
That was my '80s, along with Floyd, U2, Dire Straits, Tears for Fears, a bunch of New Wave, etc.
Many are solid candidates. Note, no effing hair bands.
"We Got The Beat" and "In Your Eyes" are probably the most iconic of that group.