Which Prophet was swallowed by a big fish?

The Prophet who was swallowed by a big fish was Yunus (Jonah, AS). This story is narrated in the Qur'an in multiple places, most notably in Surah As-Saffat (37:139-148) and Surah Al-Anbiya (21:87-88). Yunus is also known by the title Dhun-Nun (the one of the fish/whale).

Yunus (AS) was sent to the people of Nineveh, who rejected his message. Frustrated by their disbelief, he left his people without Allah's permission. He boarded a ship, and when a storm threatened to sink it, the passengers cast lots to determine who should be thrown overboard to lighten the load. The lot fell on Yunus, and he was cast into the sea, where a great fish swallowed him.

Inside the belly of the fish, Yunus (AS) called out to Allah with the famous supplication: "La ilaha illa Anta, Subhanaka, inni kuntu min adh-dhalimin" (There is no deity except You; exalted are You. Indeed, I have been of the wrongdoers) (Qur'an 21:87). Allah accepted his repentance and commanded the fish to cast him out onto the shore. He was then restored and sent back to his people, who this time accepted his message.

Musa, Ibrahim, and Nuh (AS) each had different trials but none involved being swallowed by a fish.