Probably still forgetting tonnes and bear in mind I'm from London so this will be GB and football (soccer) heavy;
Women's GB Cycling team wins Team Pursuit Gold London 2012 after breaking their own WR about 5 times
Can't find a video of the TV footage but the day in the 2012 London Olympics when Mo Farah, Jessica Ennis and Greg Rutherford all won track Golds for GB in the same night
Can't remember them as well but there's also; Kelly Holmes double Gold in Athens, Team GB wins 100M relay 2004 and Johnny Wilkinson drop-kick to win the WC for the England Rugby Team. I also have vague memories of Iconic Ronaldiniho, Zindane, Ronaldo R9 and Arsenal moments but I'd be lying if I said I remember them from the time.
Best I've seen live is Sanchez volley vs City (mainly because I was right behind the goal) and Eduardo getting a standing ovation coming back to the our stadium for a different team (for those who don't know he was a very promising player for Arsenal who had his leg broken in a match and was never quite the same).