At the beginning of the episode Jesse references his uncle Nick, which doesn't make sense because Nick is the name of his father. It is extremely unlikely that Jesse's grandparents would have named 2 of their sons (Jesse's father and uncle) Nick.
When Michelle starts to carry Martin away for a walk, he swims nears the top of his bowl. He is still there in the next shot, but in the next he floats closer to the center.
Fish don't give live birth. They lay eggs.
Joey suggested to Jesse and Danny that to prevent Michelle from knowing that Martin is dead is to put him on a little raft and tell her he's working on his tan. He could've just bought another goldfish that looks like Martin and replace her dead fish with that.