When Beckett is showing Capt. Montgomery the map of the sewers, she follows the 22nd Street sewer line with her finger. However, instead of "22nd St." the map reads "22rd St."
When Castle and Beckett go to question "Magoo", Beckett introduces herself and shows him her badge; when the shot cuts, she shows it to him a second time.
At the end, Captain Montgomery presents Castle a bottle of whisky with both hands. Castle accepts the whisky and says that he wants to share it. Captain Montgomery eagerly presents a glass to get his share of the whisky. The glass seems to have appeared from thin air as it was not in his hands when Captain Montgomery gave the whisky to Castle.
The bottle of 1875 St. Miriam shown in the auctioneer's archive photo has a label which reads "Scotch Whiskey". "Scotch" whisky is spelled "Whisky". The "e" only appears in non-Scottish Whiskeys.
The lab supposedly finds traces of alcohol on a shard of glass that was embedded in the skull of the body that had been underwater for hours. Alcohol is soluble in water and all traces of it would have washed away in the several hours the glass was underwater.
When Becket and Castle are looking at plat/sewer maps, Becket points to where the mystery man got away. Next to that, the map clearly shows The Banning Museum. The Banning Museum is in the Wilmington district of Los Angeles, not lower Manhattan. Either the show used real plat maps of L.A. (unlikely) or they made there own and added this intentionally.