During the whole series the moon is seen rising from the ocean, and that's correct because Gold Coast is on the East Coast, but the moon can even be seen setting on the ocean, which is impossible from that location, because only on the West Coast can the moon be seen setting on the ocean.
Emma, Rikki, and Cleo (and later Bella) all wish to keep their identities as mermaids secret due to various risks such as being captured for lab experiments or zoo exhibits if the wrong people find out. Yet these characters repeatedly discuss being mermaids and factors related to it, such as Mako Island, in public areas like the cafe where anyone can just walk past and hear them saying that they're mermaids.
Cleo and Emma pick two of the worst possible jobs for people who grow tails mere seconds after contact with any amount of water. At a marine park and a juice bar, respectively, where they could get wet very easily. It's a miracle that very few accidents occurred and none of the ones that did revealed their identities as mermaids.
Many times throughout the series, characters are shown using snorkels while diving, supposedly allowing them to breathe underwater. This is possible only if one stays at the surface to allow the snorkel to stick out of the water, and this is not how they are used in the show.