"Shake Your Rump" (Beastie Boys)

"Shake Your Rump" from the classic album Paul's Boutique (1989) is the line between "the old Beasties" and "the new Beasties" for me.  All you have to do is listen to the first :30 of the track, and you instantly know that the frat boy shenanigans of 1987's Licensed to Ill are a thing of the past.  In fact, they'd had a highly publicized split with Russell Simmons, Rick Rubin, and Def Jam Records in 1988, the result of the band wanting a bigger cut of the profits and more creative control.  This ultimately led to the Beasties relocating from New York to L.A. and hooking up with the production team of John King and Mike Simpson (a.k.a. The Dust Brothers).  It also spawned weird rumors that the band was defunct because Michael "Mike D." Diamond had died from an overdose.  (I can't help but wonder if Rubin and Simmons might have had a hand in spreading those very rumors.)
Anyway, Mike D. wasn't dead, as he addresses in the second verse of the song (Well I'm Mike D., and I'm back from the dead / Chillin' at the beach down at Club Med...).  And the Beasties weren't kaput.  In fact, their tag-team flows were even more intricate and filled with pop culture references than ever before: they namecheck everyone from Fred Flintstone to The Brady Bunch's Sam the Butcher on "Shake Your Rump."  
On top of that, the song introduced the world to a whole new way of sampling recorded material.  No longer were the Beasties just copping some Black Sabbath riff and looping it over a single Led Zeppelin breakbeat; the track was one, big sound collage (there are at least one dozen identified samples on the track), culled from a vast library of vinyl and pieced together in completely unexpected ways.  It was as inventive as it was funky.
My favorite moment of this song is still that first instance when DJ Hurricane scratches in Afrika Bambaataa saying Shake your rump-ah! from his 1984 collaboration with James Brown, "Unity, Pt. 2," and then segues into that fantastic Moog synth growl from Rose Royce's "6 O'Clock DJ (Let's Rock)" from the Car Wash soundtrack.  It's still as fun and fresh as it was in 1989.

"Shadrach" (Beastie Boys)

Back in 1989 when the Beastie Boys' Paul's Boutique was released, no one really knew what to make of it.  See, there were two rumors circulating at the time: 1) Michael "Mike D." Diamond had died from some kind of overdose, and 2) the followup to the group's multi-platinum License to Ill (1986) was nothing more than snippets of unfinished tracks that had been remixed and rush-released to capitalize on Diamond's untimely demise.
But Mike D. wasn't dead.  And Paul's Boutique wasn't just some odds-n-ends pastiche of recycled rhymes about frat boy buffoonery culled from demo tapes, thoughtlessly slapped on top of random beats—despite what some music critics and gossip-mongers claimed.
It was a groundbreaking exploration of sound that took sampling to new heights.
One of the key tracks on the album is "Shadrach," a stanky chunk of funk, primarily built upon a sample from Sly & The Family Stone's 1974 single "Loose Booty," that features tag-team rhyming and mile-a-minute pop culture/autobiographical references.    
The title, "Shadrach," not only refers to the chant that Sly and Freddie Stone repeat, ad infinitum, in "Loose Booty" (Shadrach, Meshach, Abednego...), it's also an allusion to the three Biblical brothers (Shadrach, Meshach, and Abednego) who were saved by an angel of God after being thrown in Babylonian King Nebuchadnezzar II's furnace for refusing to bow to an idol.
It's actually the latter that has the most to do with the Beasties' lyrics.
They were three guys who'd been vilified by the mainstream media across the globe for their (satirically) offensive rhymes and (comedically) loutish behavior.  At the same time, they had sold tons of records based on their "scream loud/drink beer/break shit" image—an image that Def Jam Records chief Russell Simmons wanted them to perpetuate to keep the cash wagon rolling.  Thing was, the Beasties were tired of playing drunken caricatures of themselves, so they broke their contract with Def Jam and Simmons's Rush Management, which led to years of legal wrangling.  Nevertheless, they gained artistic autonomy and started over in LA with producer Mario Caldato, Jr., and crate-diggers extraordinaire The Dust Brothers (John King and Mike Simpson) helping them chart a new course.
So they had been through the fire, and they had not only survived, but they'd also remained true to themselves.