As far as I know, there is no notation in a classical score asking for a mute (fancy Italian word: sordina) until well into the 20th century. After all, you choose to write for a trumpet because the thing is so damned loud. Likewise, no mutes in Sousa, or James Reese Europe.
We pick up the use of muting in full bloom with King Oliver in the early 1920's. By all accounts, he was apt to use anything that was close to hand-bottles, glasses, hats... No doubt Oliver had progenitors in New Orleans, but of recorded evidence, there is none. This may be his most famous muted solo:
People ask, who was the first person to eat a tomato?
More importantly, who was the first brave soul to use a plunger mute?
The deftest lineage of mute practitioners were the plunger players associated with Ellington: Bubber Miley in the 20's, then Cootie Williams, then Ray Nance. On the trombone side, "Tricky" Sam Nanton. Here's a famous Bubber example:
There was a school of players who used the Harmon mute in what we think of now as a ricky-ticky way-stem left in, hand moving over the end to get a wah-wah effect. Henry Busse was a well-known practitioner (one odd article claims: "He and singer Bing Crosby invented the mute for trumpet").
Here's Busse in "Hot Lips."
The Lunceford band was known for flashy choreography, which extended to flashy sectional use of mutes. It comes near the end of this 1936 compendium, but the whole thing is great, so you'll hang in:
In 1938, Tommy Dorsey used it to achieve a putatively "exotic" quality in "Song of India" (don't forget to listen to the great Bunny Berigan solo, too):
The cup mute got some traction in the 30's-40's. Here is the alt take of Bird's "Embraceable You." Miles uses cup mute in his solo:
Miles is responsible for the last significant change in mute use. He took the stem out of the harmon mute and played right up on the mic. This is the sound we most closely associate with Miles and with trumpet mute use in general. Used famously here:
Next time: Mute: Threat or Menace?