No fair using anyone who actually had some legs and made it out of the 80’s. Just the flash in the pans. Exceptions can be made for early 90’s songs with an 80’s feel to them, I suppose…
As I mentioned in the Name your “guilty pleasure” song thread, mine is Heart & Soul by T’Pau. I’m also partial to the Salieri mix of Rock Me Amadeus, by the late great Falco.
if we’re talking T’Pau, I preferred china in your hand
“Tarzan Boy” by Baltimora (R.I.P.)
“Voices Carry” by Til Tuesday – they had one more song that barely snuck into the lower reaches of the Billboard Top 40 Singles chart, but “Voices Carry” is the one song that band in known for. Bassist Aimee Mann has since had a successful solo career.
“Perfect Way” by Scritti Politti (sp?)
Not exactly one-hit wonders, but my favorite 80’s song by a band that debuted in the 80s and did not last long is:
Men At Work - Down Under
Icicle Works Birds Fly (Whisper to a scream)
The Future’s So Bright, I Gotta Wear Shades - Timbuk Three
“The Safety Dance” is pretty much the pinnacle of human culture.
Echo and the Bunnymen, with their cover of “People Are Strange”…
Take on Me by A-Ha (I hope that’s the right capitalization of the band name; I’m too lazy to look it up right now.)
“I Melt With You” by Modern English! I have always considered this the ultimate one hit wonder song, since many people I know don’t even know the name of the song, and simply call it “Modern English”.
Which I guess is better than folks knowibng the name of the song and not the groupJane’s Getting Serious by Simon Townshend (Pete’s Brother)
The Promise by When in Rome, featuring Midge Ure
There are plenty others, but those two come immediately to mind…
I was going to say that! The video also has warm, nostalgic place in my heart.
“I Want Candy” by Bow-Wow-Wow.
Did Katrina and the Waves do anything besides “Walking on Sunshine?”
Who did “Turning Japanese”? I don’t think they had any other hits.
“Puttin’ on the Ritz” … who did that?
The Vapors and Taco, respectively.
Danny Wilson Mary’s Prayer.
Although the name sounds like one guy it was a band - named after the Sinatra movie Meet Danny Wilson. The bright boy behind the band was Gary Clark. The song is a pop classic - well written, beautifully performed and with tremendous production values. Whenever I listen to the song I marvel at the fact that these guys, with all this musical talent, basically amounted to nothing.
Find a copy, put on the headphones and pay attention to all the clever little musical phrases.
Several of these are by groups who were one-hit wonders only in the U.S. and much bigger elsewhere, but that’s life.
“Goody Two-Shoes” Adam Ant
“I Wanna Be a Cowboy” Boys Don’t Cry
“Too Shy” Kajagoogoo
“88 Lines About 44 Women” The Nails
“Life in a Northern Town” The Dream Academy
“I Eat Cannibals” Total Coelo
“Under the Milky Way” The Church
BTW, virtually every song mentioned in this thread is available in the Living in Oblivion, The 80s Greatest Hits series. An absolute must for any car journey.
Wall of Voodoo : Mexican Radio
…eating barbequed iguana.
“Walking A Fine Line” by Johansen
“Life’s What You Make It” by Talk Talk
“Smoke Two Joints” by The Toyes (which I just found after looking for it for 22 years!)
“Walking On Sunshine” by Katrina & The Waves"
“Black Man Ray” by China Crisis
Come on Eileen by Dexy’s Midnight Runners
I know 95 % of of the songs mentioned here. Blame radio station CFNY–this is exactly what they used to play.
The eighties: when “alternative” meant something! 
I’ll add:
Jet Town by SSQ. Still trying to find the album on CD, vinyl, P2P, anywhere. Had a cassette way back when, but it died.
Echo Beach by Martha and the Muffins (later M+M. his group was not really a one-hit wonder, but this their most famous song. Still available on CD (I saw it last week at Sam’s).
Smiling Monarchs by the Abecedarians. And you don’t want to know how long it took me to find out how to spell their name…
Oh Yeah by Yello. It was getting airpley here well before it was used in Ferris Bueller’s Day Off.