I’m guessing you live in the U.K. The record charts in the U.S. are a different story.
Of all the artists listed so far, only Falco (2), 'Til Tuesday (2), Men at Work (5), Men Without Hats (2), A-Ha (2), Katrina and the Waves (3), Adam Ant (3), The Dream Academy (2), General Public (2), Alannah Myles (2), Swing Out Sister (2), Glass Tiger (4), Real Life (3), The J. Geils Band (10), Survivor (8), Stacey Q (2), and Boy Meets Girl (2) had more than one hit on the Billboard Top 40 chart.
Bourgeois Tagg–I Don’t Mind at All
Dexy’s Midnight Runners only had one US hit!!!?? This confirms my belief that the US is a cultural wasteland.
This is probably the '70s. I’m sure it must be.
Norman Greenbaum - Spirit in the Sky
They only had 11 hits (including two #1’s) in the UK*. Not exactly setting the world on fire, eh?
*Guiness World Records British Hit Singles 15th Edition ©2002
Lots of good ones listed above. Here’s mine:
“What About Me” by Moving Pictures (1982)
Just put it in my CD player and am listening to it now.
Also, I believe “Lunatic Fringe” was by Red Rider.
I’m frightened by how many of these songs exist in my iTunes library. 
Some of the most often mentioned “one hit wonders” of the 80s did indeed release second singles which, in many cases, were superior songs. 'Til Tuesday’s follow up “Looking Over my Shoulder” was far better, IMO, than the cloying “Voices Carry.” A-Ha hurt themselves; there was so much notice about groundbreaking rotoscope animation in the video for “Take On Me” that people burned out on it. When they went back to rotoscoping for “The Sun Always Shines on TV” people didn’t pay much attention, and the single did nothing on the charts, either. All of Swing Out Sister’s debut album “It’s Better to Travel” was terrific, it just wasn’t pop, and therein was the rub; people expecting 12 tracks like “Breakout” had to be disappointed by songs like “After Hours” which were straight up chanteuse-nightclub modern poppish jazz. They’re still together and recording. Their latest album “Where Our Love Grows” comes out on October 19 and the tracks I’ve heard sound great, and are more in that light jazzy-pop genre.
Actually it was written and originally recorded by the late, great Kirsty MacColl who was a good friend of Tracey’s. Kirsty sang backup on Tracey’s version, and the “Baby!” you hear right after the guitar solo mid-song is actually Kirsty; Tracey couldn’t pull it off successfully (to her own satisfaction) and asked Kirsty to sing it for her. The video, with a cameo from Paul McCartney, is outrageously adorable and kitschy, much like Ullman herself.
One of my favorites is “The Politics of Dancing” by a band called The Reflex who I think would’ve had a greater success if not for the confusion between their band name and the Duran Duran song of the same name.
I also always liked “Digging Your Scene” by The Blow Monkeys, but it may have been the name of the band (and the swanky video for the song) that really caught my attention.
Another favorite is a song that came out in late '89 (just skirting the fine line of 80s one-hit-wonders) called “No More Lies” by a baby-voiced singer called Michel’le. She was, at the time, engaged to a vaguely known rapper who happened to produce her record for her and showed up in the video for the song. They broke up and she faded into obscurity. 15 years later, though, everyone knows her ex, the rapper – Dr. Dre. 
Not only did they only have one US hit, no one in their right minds could say it was their best song. When I pointed out to someone that they released a 14-track greatest hits compilation in the UK, their eyes bugged out of their head.
enipla, Norman Greenbaum’s version of “Spirit in the Sky” was released in the 70s (perhaps even the 60s) but it was released by weird new-wavers Doctor and the Medics in 1985.
It was released in 1970.
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WooHoo! A fellow Fuzzbox fan! I actually came in here to say that they are my favorite no-hit wonder band. “Love is the Slug” and “Jackie” are my two favorite songs on We’ve Got A Fuzzbox…
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On-topic:
“(There’s) Always Something There To Remind Me” by Naked Eyes is one of my verymost favorite 80’s songs, and as far as I know, Naked Eyes didn’t have any other hits. But they do have a Best of… album, so I’m probably wrong.
Was it re-released in '85?
Whenever I think of one hit wonders I always think of Norman.
Norman Greenbaum.
First time I saw his picture was today. I always pictured him as a 13 year old Jewish kid playing guitar in the basment. WAY TO GO NORMAN!
“Pilot of The Airwaves” by Charlie Dore
“I Wish I Had A Girl (Who Walked Like That)” by Henry Lee Summer (he may have had another song low on the charts, something like “Heard It On The Radio”)
“Walls Come Down” by Glenn Burtnick (he may have had more, this is the only song I’ve ever heard of his, though)
Sir Rhosis
Not by Greenbaum. His version has been in a ton of movies, though, so it seems to never go away.
I read a story about him not too long ago that said he has been living off royalties of that one song for nearly 30 years.
Great harmonies in that one.
Walk the Dinosaur - Was, Not Was (video was funny as all heck, too)
Mercedes Boy - Pebbles
Punk Rock Girl - The Dead Milkmen (another hilarious video)
Man, I hated Come on Eileen. With a serious passion. I had no idea there was a greatest hits album. I always find it funny to look at the differences between the US and UK charts for the time.
Funny too how some people name a band, and I think I know what song’s coming, and they say something completely different. For the following bands, I definitely remember:
Shriekback - Nemesis
Captain Sensible - Wot
Alexi Sayle - Didn’t You Kill My Brother?
Others that come to mind:
The Timelords - Doctorin’ the Tardis
Screamin’ Blue Messiahs - I Wanna Be A Flintstone (although I like the whole disc, especially Jesus Chrysler Drives A Dodge.)
National Velvet - Shine On and Flesh Under Skin
Johnny Rotten & Afrikka Bambatta - World Destruction (I know they each had more hits, but this was their only one together!)
And then the dance tunes! Some of which may be early 90’s, but it’s my thread and I’ll darn well do as I please:
Bizarre Inc. - I’m Gonna Get You
Bomb The Bass - Beat Dis
Captain Hollywood Project - More and More and More
Man To Man - Male Stripper
S-Express - Theme From S-Express
There were other bands that had a couple of dance hits too, like 2 Unlimited, Black Box, Snap & Technotronic.
Boy, I have some downloa…, I mean, buying to do. Yessssss, buying…
OK, I’m back from Birthday Dinner with some more fave '80s one hit wonders. Let me just look through my singles… some of these will only be familiar to Canadians, as that’s where I lived at the time.
“People Who Died” - The Jim Carroll Band
“Too Bad” by Doug & The Slugs
“Steal Away” by Robbie Dupree (which might as well be called “Steal The Riff From What A Fool Believes”)
“Psycho Chicken” by The Fools (parody of “Psycho Killer”)
“Moscow Drug Club” b/w “Nyet Nyet Soviet (Soviet Jewellery)” by BB Gabor
“My Mistake” by The Kingbees
“In My Room” by Billy Thorpe
“The Friends Of Mr. Cairo” by Jon & Vangelis
“Mercury Blues” by David Lindley
“Harden My Heart” by Quarterflash (they hated that song for being so popular and ruining their career)
“Mainstreet USA” by Union
“Our House” by Madness
“Talking In Your Sleep” by The Romantics
“Break My Stride” by Matthew Wilder
“Chayla” by The Dice
“Had A Dream (Sleeping With The Enemy)” by Roger Hodgson
“Wouldn’t It Be Good” by Nick Kershaw
“If I Was” by Midge Ure
“Polka Dot Undies” by Bowser And Blue
Anybody like any of these?
I’ve got the Robbie Dupree album on vinyl. Great make-out song.
What the hell was that song that was cool when you were 16, but is rather creepy now. The one that went, “She’s just 16 years old, leave her alone, they all say”?
Sir Rhosis
That would be Benny Mardones’ “Into the Night”. He bookended the 80’s with that:
#11 in June 1980 and #20 in May 1989.
Same song, twice the creepiness.
Sixteen’s legal in Canada, the U.K., most of Europe, and at least 26 of the U.S. states.
Something About You- Level 42
Never Gonna Give You Up- Rick Astley
You Spin Me Right Round-Dead or Alive