According to a deleted scene, the shuttle was named Copernicus.  The registry number of the shuttle was somewhat illegible on screen, but appeared to be BA-30. The roof of the craft included the identification numbers DF723 on the outside and FAB.1X456.31 on the inside, though it is unlikely that either of these was the registry of the shuttle based on their locations.
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For the sequence of shuttles landing in the Enterprise main shuttlebay, Shuttle 7 can be seen on a landing approach.
For the sequence showing shuttles tumbling from their perches, the Shuttle 7 was seen twice, once in the foreground and once in the background. Shuttle 4, situated between the two, was unlabeled with a name and might have been a CGI creation.