I’ve gotta agree that it’s most likely “Don’t You Forget About Me”, or maybe “Beat It”, but I’d also like to throw The Go-Go’s “We Got The Beat” into the running.
The first thing I thought of was the scene in Pulp Fiction where Jules says to the white boy, ‘You, flock of seagulls, you know why we’re here?’ So I’m going with anything by them.
My funeral song, and for me, the most perfect pop song yet written.
Wake Me Up Before You Go-Go by Wham!
If you are unaware of the 80’s here’s the video.
MiM;););)
But on that multy parody/nostalgia angle I thought of what GuanoLad mentioned and one can not really avoid the overproduced music videos of the day and Phil Collins made a parody of all that back in the 80’s:
[Billy,] “Don’t Lose My Number” (and yes, it was released in 1985 
)
It made fun of The Police’s Every Breath you take, David Lee Roth’s California Girls, The Cars’ You may think I’m crazy and others. (And movies like Mad Max, The Road Warrior)
Eurythmics - Sweet dreams (are made of this)
Several of those already mentioned had also popped in my head (Human League, Michael Jackson, Duran Duran).
Never heard that before, I likee.
The first thing I thought of was Shout from Tears for Fears. Then I thought of Pat Benatar and Hit Me with Your Best Shot and Tom Sawyer from Rush. I really can’t pick one thing for the whole decade; I listened to a lot of different kinds of music and had a lot of fun in the '80s.
“Centerfold” is the '80s song for me. Graduated HS in '89. 8th grade '85! First glasses, 1977!
Fuck! Help me, fellow old people! 
You went up town ridin’ in your limousine with your fine Park Avenue clothes,
You had the Dom Perignon in your hand and the spoon up your nose. . .
– Big Shot, Billy Joel
My first thought. It even has a long enough gap for local radio stations to put in their own promos. It panders to every 80’s hook in existence.
99 Red Balloons
I’m surprised that the first two I thought of still haven’t been mentioned (unless I missed a video link somewhere…)
“Take My Breath Away” by… damn I don’t even know who did it. Berlin?
And Thompson Twins “Hold Me Now”.
I always wondered if that was autobiographical.
Wow, 73 posts in and not one mention of Yello’s “Oh Yeah”.
Is the OP looking for “instantly recognizable as a song from the 80s,” or “songs from the 80s that make us cringe,” or are we simply making a list of all the songs we remember from the 80s (which is inevitably what we’re going to get anyways)?
A joke I heard on the radio one time: "There’s a new song out, a mashup with Wham!, Michael Jackson, and Lionel Richie. It’s called “Wake Me Up Before You Go-Go, or I’ll Beat It, All Night Long.” 
That’s what I came in here to say, and it was done in the second post. Not that I’m fond of this song or anything, but (imho) it’s the perfect answer to your question.
ETA, I hadn’t thought of it until you posters mentioned it, but the Modern English song is a great answer, too !
It’s difficult for me to choose just one song because the 80s fall into two or perhaps even three distinct periods:
1 - Synth pop and New Wave (1980-84ish)
2 - “Bland” pop and Hair Metal (1984-88ish)
3 - Acid House (1987-1990)
My go-to example for #1 is Visage’s Fade to Grey (http://www.youtube.com/watch?v=UMPC8QJF6sI&list=RDUMPC8QJF6sI))).
#2… I don’t know. Guns N’ Roses: Welcome to the Jungle?
For #3 I’d pick Theme from S-Express by S’Express (http://www.dailymotion.com/video/x1hibc_s-express-theme-from-s-express_music). I hated it at the time. Really like it, now.
If I had to pick just one, it’d be Fade to Grey.
I’m also puzzled by all those mentions of Don’t you Forget about me. It definitely sounds like a typical 80s song but I wonder why it seems so iconic. *The Breakfast Club *effect?
The song that seems to be on the cusp of all is a different New Order track - Blue Monday. Synth, guitars, pretentious lyrics, and if you don’t dance to this, you’re dead (or undead mike-pole-beating Freddy Mercury).
Guns N’ Roses? BLAND? It is pistols at dawn, sir.
A better (much better) example of #2 would be something like this. Or any of a dozen more songs/acts.
Sorry. I was in a hurry last evening and didn’t supply the links (I’m sure you - and most other people - have heard at least the first two before).