Black Rock Shooter's opening theme is sung by Miku Hatsune.
Are there any other anime soundtracks that incorporate a vocaloid singer (or a similar synthesized voice program) such as Miku?
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It's probably going to be pretty rare for anime soundtracks to be featured with a Vocaloid singer, since the talent tie-ins and money for making anime want to promote their in-house talent so they want to use their own (live) singers. You can see a lot of famous vocaloid producers get hired for their music but instead of the vocaloid voice track, they use in-house talent for the voice:
Jon Lin's answer is good, so I don't think I have anything to add in that direction. As he says, using Vocaloid singers for songs of full-length anime (rather than animated music videos) is rather uncommon. It sometimes happens that a Vocaloid singer will be used in early stages of production, but by the time they are ready to release the song it's usually done by a live singer, which is generally more profitable.
I only know of two instances of Vocaloids being used other than in the Black Rock Shooter shows (which you mentioned). Both of these were atypical.
The ending of episode 13 of Akikan! (the final episode) used Miku (as Mikkuchu Juuchu). However, each episode of Akikan! had the same song (Koisora Recycling) sung in a different arrangement by Nomiko, sometimes with guest artists, so Miku wasn't really the focus.
Meguriau Sekai, a 15-minute low-budget independent anime, has an insert song by Miku called Wish: Kutsuhimo o Musunde. I was not able to find this song anywhere, though if you can find the anime it plays from 12:00 until the end. This was very likely done to keep costs low. The anime only has one voice actor, and licensing a professional musician would probably have been beyond their budget.
I searched for any other anime with Vocaloid songs, but I was not able to find any for the top 10 or so most popular Vocaloids. If there are any others, they are low-budget independent productions like Meguriau Sekai, since a high budget show using Vocaloid songs would probably be notable enough that it would be documented in many places.
It's considered as the first anime movie to use Vocaloid in its OST.
The last track, "The Girl in Byakkoya", features three spoken lines in Vietnamese. The soundtrack is significant for being the first made for a film to use a Vocaloid (using the Lola voicebank) for some vocals.
You can listen to the song and download it freely on Susumu's official website.
Episode 6 of Watashi ga Motenai no wa Dou Kangaetemo Omaera ga Warui was sung by Miku, but it's kinda sad that it was only for an episode. And it seems like it wasn't even an original song, I think, but just a cover. Not sure, though.
Black Rock Shooter, in fact isnt the only OP sung by Hatsune Miku. Heat Haze Days, from Mekaku City Actors is also sung by Miku.