Continuity mistake: When Richard Denning and Richard Carlson approach the boat ladder coming back from their first dive in the Black Lagoon, you can see the net hanging underwater just to their right. It already has the huge torn hole that is in it when they hoist it up after the creature is caught for a moment in a later scene.
Continuity mistake: When the scientists find the creature climbing onto the ship at one point, it's night time. It smashes a lamp, then jumps back in the water, swims across the lagoon, then standing up near the water's edge, it grunts and slashes at the spotlight they aim at it, then goes back underwater. Two of the scientists dive in to go after it, and follow it through an underwater (then above-water) cave. When both creature and scientists exit the cave, however, it is not only daytime, but the other scientists are waiting on the shore nearby.
Plot hole: The creature gets caught in the fishing net and (after a struggle that shook the whole boat) freed itself. The scientists, upon drawing up the net, find one of the creature's claws, and thus they now seek its source. Shortly after, while Mark is busy loading a harpoon gun, Dave is protesting, stating, "This...this thing *alive* and in it's natural habitat..." It would seem that at this point, they know the creature is *not* dead, but very much alive. They dive, they see it, Mark shoots it with a harpoon, and then upon surfacing, they argue a bit before reporting what they saw to everyone else. So why after seeing it underwater, fully knowing that "this thing (is) alive," do they act so surprised to see it, as if they didn't have a clue it could possibly still exist?