Yes.
Many times.
Just google "B/W Same" (which means "Backed With Same") and you'll find many images of singles marked this way.
An example of this can be found here .
Singles were typically released and "backed with" a track that was a strong contender for airplay.
As music moved into the 70's, singles would often get backed with alternate tracks that didn't appear on an album, as to make fans more likely to buy them.
Some famous 45's like that were "
Hey, Hey, What Could I Do" by Led Zeppelin, and "Abilene" by Yes.
musicfans.stackexchange.com/users/53/Johnny Bones
One of the most unusual cases is the song American Pie , by Don McLean.
The album version is listed as being 8:33, making it the longest number 1 song in history, but at that length, it took up both sides of a standard single.
The length part of the Wikipedia entry list the length of side A as 4:11, and side B as 4:31.
musicfans.stackexchange.com/users/27/Donald.McLean
no, singles had 2 sides, (A-side and B-side)
so there would be at least 2 songs.
I say "at least" because some B-sides might have 2 songs.
And some B-sides have become famous in their own right - Queen's
"We will Rock you", for example.
musicfans.stackexchange.com/users/65/Angst
I've seen also singles where in the A side is the original song and on the B side is an instrumental version.
Another thing could be, that on the A side is a cover version or a remix and on the b side is the original one.
For example: "Swing the mood" by Jive Bunny and the Mastermixers.
On the A side is the radio version of Swing the Mood and on the B side is the original one by Glenn Miller Medley ( http://www.discogs.com/Jive-Bunny-And-The-Mastermixers-Swing-The-Mood/release/541208 )
musicfans.stackexchange.com/users/124/LStrike
Let's say 'nothing famous' was done as a single track, but Google has a few suggestions for '1-sided vinyl' including one interesting hit of a release that was CD on one side, record on the other.
There's an entire wikipedia article on A-Sides & B-Sides
musicfans.stackexchange.com/users/84/Tetsujin
This question is fraught with conflation.
Everyone here seems to be trying to shoehorn answers to fit with the original question.
When the term came about, 'single' was not intended to connote that you would only be buying one song.
Music was first distributed on vinyl records.
You could purchase vinyl in three options: LP, or Long-Playing.
This vinyl record consisted of roughly 5 songs per side.
It was spun on a record player at 33 1/3 revolutions per minute (RPM) EP, or Extended-Play.
This vinyl consisted of a number of total songs less than an LP, but more than a single.
Single.
The intent on releasing a single was putting the best foot forward to promote an entire album, or LP.
Singles were only a song in length on each side.
Because there was less content, it was recorded differently (to take up more vinyl real estate), and ran at 45 RPM.
Some LPs had strength enough to release multiple singles.
(Michael Jackson had a number of LPs where this was true) The single, or reason that this single song was released, was consider the A side.
The B side almost always contained a weaker (and different) song, used to draw interest on the rest of the album.
Over time, artists would put songs on the B side that would never be contained on a studio LP, to generate rarity and more reason to purchase singles.
musicfans.stackexchange.com/users/962/Jason P Sallinger
Every single I have on vinyl, cassette, and CD has at least two tracks, even if they are alternate mixes of the same song.   With downloads you can obviously buy individual songs these days, but before that, when you bought a single, did you ever really get just one song?
Before the days of downloads, were 'single' releases ever really only one song?
musicfans.stackexchange.com
<recording>
musicfans.stackexchange.com/questions/248
musicfans.stackexchange.com/users/None/
Singles had two sides, and A and B side. You could get a single but you would never really get just one song seeing as even singles had two sides. 
Yes No, because singles had two sides