Is there a computer program that I can use where I upload a favourite song and it'll transcribe it to sheet music for me? I'd like to play Running up That Hill by Placebo but I don't want to buy the sheet music online.
Simply put, no.
If you limit a song to single pure tones, it's pretty easy to write software to recognize them and transcribe it. But once you get to a real instrument things get much harder. Even single notes can be difficult to recognize due to overtones -- the dominant frequency doesn't even need to be the fundamental frequency, which makes it very difficult for software to get it right.
Then add in other instruments and chords and effects and what have you and the problem becomes incredibly hard. There is no software in the world that you can feed arbitrary songs and get an accurate transcription out every time (or even a meaningful percentage of the time for non-trivial songs).
Your best bet is human transcription, possibly coupled with audio editing or transcription aid software to help you out, e.g. Seventhstring's Transcribe! software (note, I haven't used it). While we don't provide direct help with transcription, we have several well-answered questions on the site regarding ear training and transcription; I'd encourage you to check them out and take a stab at learning this song yourself!
There is no route:
song => sheet music
But there are routes
song => human "translator" => midi file => sheet music
There are tons and tons of "midi files" all over the internet. You only need to search Running up that hill midi (that one's by Kate Bush, I'm not sure that's the same as the Placebo version)
Synthesia can display sheet music from a midi (see video on site)
For most songs it's more trouble than it's worth when not impossible.
If you only want the piano section for this song, it might be possible, but there's not one app that will do this. You first need to find a midi version of the song instead of wav or mp3 and a midi player that will let you turn off channels (vocals, bass, drums, etc) then save the results. The next step is to find an app that will convert midi to notation (reliability varies, but the piano on this song is pretty simple so you may have good luck).
There are a few software titles that do this, I don't have a specific recommendation since this is usually an exercise in futility for more complex pieces... I tend to not go this route. The ones I've tried also did TAB, and those tend to yield impossible results. If you want the keyboard and synthesizer sections you probably won't have much luck transcribing with the effects they're run through, but the piano in this song does not seem to have very many polytones that notoriously botch most software transcriptions.
Still, at the end of the day, it'll probably be less frustrating just to buy the sheet-music.
If you have a midi version of the song, it's pretty much done for you. You can open it in a program like guitar-pro and it will transcribe sheet music and a tab for you. I don't know how useful the tab is to you, but you get the former out of it as well so.
You may try http://codingteam.net/project/scolily unfortunately you need Linux to use it.