When Ryan gets hit by the black taxi, his shoe comes flying off. Then it appears back on his foot in the next shot.
When Falcon and Valo find the poster for the "Invention of the Future" contest, they are standing next to an early-1990s dark gray Nissan Maxima. In the next shot they are standing alongside a late model gold Ford Crown Victoria/Mercury Grand Marquis, but if you look down the street, you can clearly see the Maxima and the posters from the original spot. By the end of the sequence they are back next to the Maxima.
In the beginning Ryan drinks from a bottle of wine. Two shots after that he has a bottle of Budweiser in his hand and not a bottle of wine.
When Falcone and Valo attempt to escape Glauren's house when Hellboy is having sex with her, the camera they are using to film the act has a "fish-eye" lens on it. After escaping from the pursuing Hellboy, Valo begins to film Falcone "getting his ass kicked" and the camera no longer has the fish-eye attached. At no point would Valo have had time to remove it.
When Falcone and Valo are getting ready to break into Glauren's house and Falcone is talking to the "Tetris Addict", during a seemingly uninterrupted series of dialog between the characters, the level and amount of tetris blocks changes many times on the TV.
When Valo and Falcone break into Glauren's house, and are in her bathroom, the cameraman is visible in the mirror in the shots taken with Falcone's camera.
In the scene where Falcone is filming/directing the toys, you can clearly see the Rake dummy used later in the film in the background of some of the shots.
When Ryan is arrested, Valo offers a sports watch as a bribe, and you can see Valo's microphone pack and wire clearly in his back pocket.