In the first shot of the Chelsea Arms apartments, the neon 's' in 'Chelsea Arms' is burnt out with an 'a' and an 'r' flickering. In a following scene, the 's' is functioning.
How did the Batcomputer have Simon Trent's fingerprints on file? Trent was not revealed to have a criminal record, so there is no rational explanation as to how his fingerprints were in any police database that the Batcomputer could access.