I never recognized any direct rip-offs, er, homages other than The Big Lebowski being similar to The Big Sleep, but the first time I watched it I recognized that it was a send-up of 40s detective noir.

Most prominent for me are the the introductions to TBL, Maude, and Treehorn, where they start conversations by talking about things that seem out of left field, totally irrelevant, and eventually come around to turning those topics into some kind of vague connection to or metaphor for the case. The fun part is, where Sam Spade or Philo Vance or Nick Charles might have followed the conversation and understood the metaphors, Dude is totally lost from word one.

Dude constantly getting kidnapped, invited, summoned, and roughed up by various players in the story is also a trope in detective noir, as is the femme fatale, the missing girl, and the weaselly rival (brother shamus) trying to horn in on the deal. Though, in noir, the weaselly rival usually ends up dead because he bit off more than he could chew.

Hell, by the end Dude himself starts to think of himself like a detective, working shit out and referring to what's going on as "this case".