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.
The best clarinetist in the world is a bit subjective. However, some of the more well-known and famous players include Anton Stadler, Johann Simon Hermstedt, Heinrich Baermann, and Jack Brymer. Anton Stadler is well-noted because he inspired Mozart to write several pieces. Johann Simon Hermstedt, although not having as beautiful of a tone, had technical ability when playing the instrument.
Heinrich Baermann was noted for his work with composer Carl Maria von Weber. This clarinetist had audience members writing the king in his praise. Jack Brymer is more of a recent prodigy and has recorded many important pieces with orchestras.