When Jake and Joey are sitting at the kitchen table, Jake tells Joey to grab a towel to hit him. When Joey picks it up Jake says "Now wrap it around your hand," but the first half of the sentence seen from behind Jake focusing on Joey, Jake's mouth is not moving.
In the entrance to the last fight as Jake makes his way to the ring you can hear the ring announcer but as Jake passes the bottom of the ring the announcer has his hand by his side holding the microphone, therefore not able to be broadcasting.
Early on in the movie, Jake tells Joey to punch him. Joey does so, repeatedly, and leaves his ring on thereby cutting Jake with each new punch. Yet right before he throws the last punch, all of the puncture wounds in Jake's forehead have been removed.
As Jake hurries to the bathroom after pouring ice water down his shorts, the top of a crewmember's head appears briefly in the lower left-hand side of the screen. (This may be peculiar to the 1988 MGM/UA videotape version.)
In a shot that introduces Jake's post-boxing life in Florida, the license tags of his two Cadillacs are shown in close-up. One of the Florida plates bears a "5" prefix, which in the 1950s indicated an automobile registered in Polk County (Lakeland). If Jake was living in the Miami area, it's far more likely his automobiles were registered in Dade County (prefix "1") or perhaps Broward ("10"), but not in a county 150 miles away.
The goof items below may give away important plot points.
When Vicki is sitting in the car outside Jake's club, telling him she's divorcing him, Jake, standing outside the car, puts a lit cigar in his mouth. The shot immediately switches to a point-of-view from inside the car, and Jake's mouth is empty. He then puts an unlit cigar in his mouth. The POV changes to back outside the car, then back to inside the car. Now Jake's cigar is lit, although he never lit it.
Jake LaMotta is portrayed deliberately throwing a fight against the "still undefeated" Billy Fox on November 1947. Fox had actually lost one professional match only a few months before: on February 1947 against Gus Lesnevich (by TKO at 10th round).