I'm a younger fan - I started following the Dolphins in 2004. We haven't been great during the time I've been following the team, but I'll tell you a few I remember:
2015 vs Houston - attended the game, went with a group from FIU. Got completely soaked. Witnessed the best half of Dolphins football I had ever seen. ****ing blast.
2005 vs Buffalo - watched on TV, Sage Rosenfels and Chris Chambers lead an epic 4th quarter comeback, with Chambers catching the game winning TD at the very end of the game.
2004 vs New England - watched on TV, highpoint of a low season, defeating the Patriots on a 4th down TD throw from Feeley to Thompson, giving the Patriots their only blemish on an otherwise successful Super Bowl season.
2008 vs San Francisco - attended the game, Dolphins controlled their destiny in the AFC East race. Boring game score-wise but a win, with Joey Porter sacking Alex Smith or whoever was quarterbacking the 49ers to end the game. Most I've ever seen the stadium full - packed to the walls. Optimism very high back then. Next week they would beat the Jets in New York to win the AFC East.
2008 @ New England - watched on TV, the Wildcat game. Blew the doors off of New England. One of the few times I've ever seen Bill Belichick outcoached. Masterful job by David Lee, and Ronnie Brown for running the offense that day.
2007 vs Baltimore - watched on TV, Greg Camarillo saves the Dolphins from certain infamy by taking a short slant from Cleo Lemon to the house for the game winning TD.
2006 vs Minnesota - attended the game, watched Jason Taylor ruin whoever was quarterbacking the Vikings at the time by having a two sack, two interception, two touchdown game...or something along those lines.