They are many skilled and excellent clarinetists in the world. It would be hard to pick one as the best clarinetist but it will be easier to determine the all time popular clarinetists.
Heinrich Baermann, played the oboe before he started playing the clarinet. He was a famous clarinetist, he travelled around Europe on concerts.
Anto Stadler is also a well know clarinetist, Amadeus Mozart wrote most of his popular composition.
Carl Baermann was a popular player, he wront a clarinet book in his time, the book is still used by many students today.
Richard Muhlfeld was amazing in his time. Here are some popular modern day clarinetists: Sabine Meyer, she is considered the most popular modern day clarinetists, Michael Collins, Richard Stolzman, Sharon kam. The list is endless as new clarinetists are coming up.