Birdo has the ability to shoot Birdo eggs as she usually does in the Mario games. When she does so, however, there is no special attack name, and it is just a basic move.
Birdo can use her snout as a vacuum to swallow enemies. There is no special name for this ability, either.
Ribbon Wrapper is Birdo's main attack in the first season, however she does still continue to use it in future seasons. It is first used in episode 04, "Witch's Revival".
To perform the move, Birdo removes and unravels her bow. She then lashes it out in a similar manner to a rope or a chain. She can use it to grab ledges, branches, or any area that will allow her to go higher or save herself from falling.
Birdo can also Ribbon Wrapper it to pull in friends that are in danger. For example, in episode 07, "Intense Revival", she saves Yoshi from Dino Piranha by grabbing him with her ribbon and pulling him to safety.
While the move is mostly used defensively, Ribbon Return can also be used as an attack as well. In one episode, Birdo wraps her bow around a giant boulder, and, with a lot of strength, swings the ribbon-wrapped boulder to attack a foe.
Unnamed Angel Wing Attack
Concept art reveals that in a future season, Birdo will develop a stronger attack/power involving her growing angel wings and then attacking the enemy with them.