"Here Comes The Weekend" was scheduled to be the lead single from The Truth about Love, but then "Blow Me (One Last Kiss)" was written and instead released.
"Heartbreak Down" was supposed to be the 3rd single from Greatest Hits...So Far!!! in May/June 2011, and actually was released in a few small European countries as a promo single, but it's overall release was cancelled. Shame, considering with a funny and cute video concept, it could've been her 3rd consecutive top 5 single. IIRC, it peaked in the mid-60s on pop radio around June, so it definitely had potential.
"I'm Not Dead" was supposed to be released as the 7th (8th if you count "Cuz I Can") single from it's title album in early 2008, but P!nk instead chose to work more on Funhouse.
Both "Catch Me While I'm Sleeping" and "Humble Neighborhoods" were released as promo singles around the time of Try This's release, suggesting that they were being considered for possible releases. However, in the end, neither were released as singles.
"Boys Boys Boys" was supposed to be released as the 6th single from The Fame, but it's release was scrapped in favor of "Bad Romance" and it's parent album The Fame Monster.
"Dance In The Dark" was favored the be the 3rd single from The Fame Monster by Gaga's label Interscope, but Gaga fought for "Alejandro" to be released, and "Alejandro" was released as the 3rd single. "Dance In The Dark" was then released as a promo single in France and Australia, but remained unreleased in the end. My guess is that they wanted to prevent Gaga from overexposure ("Alejandro" became her first single to not peak at #1 on pop radio), and to build up hype for "Born This Way" (radio-wise, Gaga had been on a 7-month drought by the time "Born This Way" was released).
"Hair" was mentioned as a possible single choice during the Born This Way era, including one radio interview where she revealed that it would be the 4th single after "Marry The Night". Also, it was mentioned by Billboard that it was considered to be the 5th single in the US instead of "Marry The Night".
"Scheiße" was mentioned to be a future single by Gaga during the commercial break of some UK TV show she was being interviewed on (I think it was Paul O'Grady or Jonathan Ross). Also, it won the GagaDaily poll to be released as the 4th single, but "Yoü and I" was chosen instead.