Despite being highly experimental, filmmaker David Lynch has had a far-reaching impact on not only films and television, but other forms of art as well, such as music and video games.
One of the most well-known directors often cited as being influenced by Lynch is Quentin Tarantino. The most commonly used examples are Blue Velvet and Wild at Heart and this isn't surprising as they portray violent criminals having trivial conversations. However, Tarantino is clearly a very different filmmaker and I definitely can't see him making something like Mulholland Drive or Inland Empire.
Tarantino's work has a very different tone and is much more meta and referential, and the Lynchian elements were more prominent in his early films like Reservoir Dogs, Pulp Fiction, and True Romance. He was most notably connected to Lynch by one of my favorite authors, David Foster Wallace.
Wallace considered the cutting off of the ear scene in Reservoir Dogs as a blatant reference to the severed ear of Blue Velvet. He also referred to the "long, self-consciously mundane dialogues on pork, foot massages, TV pilots, etc. that punctuate Pulp Fiction's violence" as textbook Lynch, as well as the "creepy/comic stylization" of the violence. He even went as far as to say that Tarantino's films wouldn't exist without David Lynch.
I think he might be overselling the connection a bit here, and I don't know that I agree with his characterization of Marcellus' neck bandage as Lynchian. But this article was written in 1996 and the work of the two directors has diverged strongly since then.