Great fish

Sperm whale ... not the Great Fish of Jonah
Image cetacenalliance

The story in the Book of Jonah is so powerful that when someone says "Jonah" this verbal sign immediately raises in many listeners minds verbal sign "fish" - assuming they are English speakers and "whale" (latin cetus) if they have pondered Vulgata or herd the urban legend of a sailor surviving in sperm whales belly.

Hebrew speakers would think verbal symbols  דג גדול "dag gadol" which means literally "big fish". Diligent yeshiva student hearing "Jonah ben Amittai" might start to think "daga" feminine form that suddenly appears in Jonah 2:1 instead of the masculine form used elsewhere.  He might ponder the rabbinical teaching that "Jonah was comfortable in the belly of the big male fish so God put him in the smaller belly of female fish so he became uncomfortable and began to pray for help". (wikipedia)

From inside the fish Jonah prayed to the Lord his God. 

Jonah 2:1 NIV

Ichthyologists might ponder that there is really no known Mediterranean fish that could swallow a grown-up man. Because of their anatomy dolphins are out of the question even the Turks call them yunus baligi, "Jonah fish" (wikipedia). Whales have very narrow throats made for eating plankton and such stuff.  It is true that complete body of man has been found in the belly of great white shark - a very dead man that is.
http://www.cetaceanalliance.org/species.htm

On closer examination modern students of the Book of Jonah would be impressed by the claim said to appear in a National Geographic 1992 volume that the Big Fish could have been a whale shark (rincodon typhus).

Whale of a tale

Credits
Whale Shark ISBN-10 : 1429617314

The whale shark is the largest of all fishes. It is not a whale (mammal). First discovered in 1828, they remained a mystery for some time. Whale sharks grow 45 ft. long and perhaps even longer. They may weigh as much as 15 tons. It has a wide, flat head, a rounded snout, small eyes, 5 large gill slits, 2 dorsal fins (on the back) and 2 pectoral fins (on the sides). It has distinctive light-yellow markings (random stripes and dots) on its very thick dark grey skin. Its skin is up to 4 inches thick. There are three promient ridges running along each side of the shark's body. But the most interesting thing about the whale shark is its huge mouth which can be up to 4 feet wide. Its mouth is at the very front of its head (not on the underside of the head like in most sharks). It has about 3,000 tiny teeth but are of little use and are covered by a flap of skin. (www.enchantedlearning.com)

But the most interesting fact of all, is the whale shark's ability to eject large objects by turning its stomach out of its mouth. (National Geographic, December 1992, Whale Sharks, Gentle Monsters of the Deep, page 127). Jonah 2:10 says, "And the LORD spake unto the fish, and it vomited out Jonah upon the dry land."
Whale of a tale

Dr. E. W. Gudger (1866-1956) was an Honorary Associate in Ichthyology at the American Museum of Natural History and a leading expert on whale sharks. He noted that "while the mouth is cavernous, the throat itself is only four inches wide and has a sharp elbow or bend behind the opening. This gullet would not permit the passage of a man's arm... The whale shark is not the fish that swallowed Jonah."
wikipedia

Important conclusion
As far as specialists on the sea creatures living in the Mediterranean Sea are concerned there is no clear candidate for the Big Fish of Jonah. Actually, it seems that literal interpretation of the story taking it as a historical description of a man being swallowed, surviving inside a big fish and being vomited on dry land appears quite unlikely to be true.

In fact, impossible in light of our current knowledge.