In the intro to the first episode, the fuse in the large Danger Mouse sign isn't lit, but the sign still explodes. The mistake is corrected in the second episode.
While Colonel K. is speaking to Danger Mouse on the video monitor, the nameplate on his desk says "Colonel K.", but in a closeup shot it says "Colonel'K".
The narrator states that DM's flat is in a quiet corner of Mayfair, but his car exits near a sign saying Baker Street, which is in Marleybone.