Since the majority of these were animated, it seemed logical to put this thread here...
Some of my favorites:
Calling Cades (1986) (http://www.youtube.com/watch?v=PhMU0LQyxEI) - Arguably the most famous Nickelodeon ident ever. If you've never seen this you either didn't have cable during your childhood or never had a childhood period.
Waiter (1985) (http://www.youtube.com/watch?v=a50OvCBLxIY) - Another durable classic that lasted well into the early-2000s (most 80s bumps got retired by 1993, this and Calling Cades were amongst the few survivors).
Disco Dogs (1993) (http://www.youtube.com/watch?v=aelFq4Dm2Ws) - Nick introduced a new visual package in 1993 that was based around their fagous "Top of the Hour" jingle from 1984. This takes a simple concept, disco music and somewhat choppily animated dogs, and makes it endearing.
Rock Concert (1993) (http://www.youtube.com/watch?v=CMytD7H2t1k) - Another one from the Top of the Hour era. Surprisingly that ditty seemed to lend itself out to all sorts of arrangements and genres.
Lincoln (1988) (http://www.youtube.com/watch?v=QIjZK8E5NgM) - Abe Lincoln gets a terrible haircut from the sinister barbershop trio of Ben Franklin, George Washington, and Ulysses S. Grant. Features an early appearance of the Nickelodeon blimp logo.
Top of the Hour (1985) (http://www.youtube.com/watch?v=cDix4ZAaZl4&feature=related) - Used at the start of every hour of programming from 1985 to about mid 1987. Replaced with "Top of the Hour II" aka Control Room in 1987.
Ants (1995) (http://www.youtube.com/watch?v=L_7yzz6Vxg8) - This is another classic. A lazy jam featuring a bunch of well fed ants at a picnic.
Fireworks (1985) (http://www.youtube.com/watch?v=34MrGPM_xlw&feature=related) - You should recognize this from the credits of Super Sloppy Double Dare.
Here's an old bump that's a sure sign of the times. (http://www.youtube.com/watch?v=lXBnygmw8D0&feature=related)
This would never happen nowadays. Never.
http://www.youtube.com/watch?v=aQ79AqEtHXo
So what's the story behind the Michigan J. Frog look a like that was featured in some of these bumpers? Even though it was at a time when barely anyone remembered who the frog was and of course it was years before the WB network started using the character, it's always piqued my curiosity how Nick got away with using him.
Nobody likes this vintage promo but I don't care, it's one of my favorites:
http://www.youtube.com/watch?v=wdhQKrRw9kA
People don't like that one? It's a classic!
^ Yeah it apparently disturbs some people. Not really sure why.
I remember a lot of those old bumpers, they definitely take me back. Hard to not wish it was the 90s again when I look at those.