For Starters
Now that you know some background information about the Chalumeau and Mr. Denner, it's time to learn some history on the clarinet itself. The first clarinets where very simple, most having only two or three keys. It was not long before many composers had started to use the clarinet in some of their pieces. In 1748 when Handel was composing Ouverture he demanded that the clarinet be included. Then in 1751 Bach introduced the clarinet to composers in London, England. It is believed that there may have been more people who initially invented the clarinet, but as there is no proof the sole credit goes to Mr. J.C. Denner.
Iwan Muller's Key System
Born in 1786, Iwan Muller was both a European inventor and a clarinettist. At the time the clarinet only had about five keys which produced the bare minimum of notes. Due to the leather 'pads' on the keys leaking air, it was hard to stay in tune and extremely hard to play accidentals (Sharps and Flats). So he decided to change this. He made the pads stuffed, whereas before they where not. This greatly improved the quality of the clarinet, and allowed him to add more keys. Thus made the ability to play accidentals without the hassle of very complicated fingerings. Muller also created the ligatures that we know today. You see before he made the ligature, clarinettists would tie their reeds to the clarinet using twine or a wire. In 1812 he presented his new clarinet to the Paris Conservatoire, where they were not impressed by it. Though come the 19th century Muller's clarinet had become vastly popular.
The Boehm Key System
Boehm later took Muller's design and yet again made it better. Using the same concept that he had used for the flute, Boehm had made a ring that when pressed down and covered with the player's fingers the ring would provide a secure seal. This also allowed him to like on the flute, attach two keys together, while needing only the one key to press down on both. The format we know today on the clarinet is the responsibility of Boehm, There has however been others that have yet again tried to improve Boehm's key system for the clarinet.