Ouch... This one will be mostly subjective and based on my emotions of said stages or the amount of impact the gave on me. It doesn't mean they were dull or something, otherwise they wouldn't produce such emotions. I've propably missed a lot of stages that i didn't watch or just plain forgotten. I will limit myself only to GT's as there was much more nice stages in smaller races but i didn't watched them or just forgotten about them.
Giro:
Sestriere 2005. I absolutely hate this stage with passion as my man Rujano (Trollo!!) just missed a GT win.
Briançon 2007. Il Cobra, Piepo and Gibo blasted open like Gewiss on Agnel (superb climb by the way) leaving like 6 or 7 other Giro condenders with them and dropping then veteran maglia rosa Andrea Noe. Then on Izoard Garzelli dropped and it came as a 5 man sprint up cobbled side outside Briançon centre leaving guys like Cobra (who cracked after blasting on Agnel), Pelizotti and Falco for over 2 minutes. It was not only like last 70-80km of tension and pressure but then my man and my bet for a GT win Andy did really great. Sadly this was one of his last important races without his brother.
Fedaia 2008. Most of mountain stages in Giro 2008 which was in my opinion more crazy than 2010 edition. The tension between Di Luca, Ricco, Sella and then quiet Contador was nerve-wrecking. While ojectively i should go with Monte Pora stage i will actually go with Fedaia one because of those last 3 or 4km on like 10% avg which lasted eternity. Ricco trying to get a gap with di Luca while Contador and Menchov were yoyoing in no man's land. Somehow this mano-a-mano gave me more thrills than the Pora stage. Other special mention would go to the Kronplatz TT which was only slightly less impactful on me than Fedaia stage.
Montalcino 2010. This one is mostly thanks to the weather. Evans in rainbow, Vino and of all people Cunego (which i call "Che Guevara") showing real flahute skills of old. Guys like Godefroot, Kelly, de Vlaeminck, van Springel or Maartens would be proud of them. Interestingly future all-rounder Nibali lost 2 minutes. Times changes as then he was more known as possible future Savoldelli but his climbing seem to got better while he now rarely shows descending skills. I thankfully didn't watched rest of this Giro because i would most propably lost rest of my hair. I never was a fan of Basso and him doing Robo just when Arroyo had a chance to do Pereiro... I don't even want to think about that. Wait those must be only mountain stages... Sheep.
Grossglockner 2011. I could get with Gardecchia 2011 but Bruce Lee was in such control that those (as allways back then) useless Nibali attacks didn't even get attention of me. But his yoyoing like Vino 2003 l'Alpe was quite funny. From 2011 i've chosen the Grossglockner stage only because of my preferences as i'm both big fan of Rujano and Contador and it was funny to see the "chasing" group watching each other.
Stelvio 2012. Blowup on Mortirollo then hestiation in the Valley as de Gent (then in top 10) was just dropping Cunego in the breakaway. De Gent's solo up Stelvio was really a great feat and Hesjedal was the only one from leader's group who showed some balls and decided to go a la Evans dragging rest of the group behind him. De Gent's solo made him a podium placing in GT while passive showing from Purito denided him his first GT win. From other stages i would propably shout-out Rabottini's solo win in rainy Pian dei Resinelli and Huzarski's 2nd in Assisi.
Verbania 2015. This Giro was sick. Astana challenging US Postal, leadership tension between Aru and sudden Landa plus shaky form of Fabio made it a really great race. Even the last stage was won by a breakaway which is a rarity these times. It was a hard time between La Spezia, San Giorgio Del Sannio, Aprica, Sestriere and Verbania but i went with the last one because this stage is an iconic example Contador's style of racing. I know it was after Landa was involved in a crash but i think even without that crash Contador would have a go. Besides it could be interpreted as a payback after Aprica.
Tour:
Luz Ardinen 2003. Propably the best stage of the 2003 edition. It was the first race (besides a particular local one which i've watched since 2000) i've watched and it made me into cycling. The rivarly between Beloki, Ullrich, Armstrong and shocking Vinokurov was tense and very memorable. This rivarly was what made me into cycling. Bust in last couple of years i've kind of lost some interest in it in favor of designing which is for me more interesting. Mayo's antics were interesting too and of course this edition was the first installment of Haimaru-sensei saga.
Mulhouse 2005. This one is technically a medioum mountain stage but this amazing solo of Chicken needs to be mentioned in this thread. Besides Dario Cioni was in the break with Rasmussen and Voigt get the yellow while Moreau (then still a GT threat) went with Voigt to gain 3 minutes (!) on the bunch.
Courchevel 2005. This one is because of two guys - Valve and Chicken. Besides it's allways nice to see Jan the Beast trying to snatch Kloeden for breakfast. This face, those teeth... Crazy times, Chavanel and Jackshe vere considered as future GT contenders while McGee was a leader for FDJ.
Aubisque 2007. Sastre trying and failing a long range (he loved going from far but most of the time with less success and than Contador). Contador trying over and over but Rasmussen keeping his cool. Leipheimer yoyoing around them and sudden bonk by Contador in the last km. Next time he would be clearly dropped in Tour would be in 2011. It were very tense last 5-10kms.
All of the Alpine stages in 2011 thanks to Contador and sudden outburst of Andy in Galibier stage. I think objectively Galibier stage should have a slight edge here because it was a definition of amazing team tacticts.
Interstingly i didn't find that many Tour stages interesting. Thankfully i've missed most of 2006 edition as i would propably kill myself courtesy of Landis. 2012 to 2015 editions didn't generate much of interest for me in mountains department. Sky and Nibali were killing them while any short-lived long range tries were too anemic to hold. Hope 2016 edition will be better in this department.
Vuelta:
This one is actually hard. First edition i've watched was in 2007. I prefer to forget the 2010 edition even if it had a lot of tension a change in leadership. Sadly my man Ezequiel lost. I somehow don't remember much from Vuelta, so think i would go with only two stages here and those are so known that they doesn't need any introduction.
Fuente De 2012 - propably most emotional and most succesful Contador trademark racing plus sneaky and very intelligent Valve and Cercedilla 2015 - Astana putting hammer down on shocking Dumoulin.
Actually i could go with Lagos de Covadonga 2014 because of Contador having problems with Purito and Valve. He just wanted to drop Froome once and for all while those two prefered to hang out on Alberto's wheel and then outsprint him gaining 3 to 5 sec. Other great factor was Froome's style of racing in this Vuelta. He raced most of this Vuelta like Ullrich on crack yoyoing back and forth, dropping and then coming back and even attacking (La Camperona). I think it was in this particular stage when he had his head down so much that he almost rode into Dani Moreno which was just hilarious. I don't remember correctly but i think it was after this stage when Contador gave his opinion on Valve's and Purito's wheelsucking behaviour and concerns about Froome who then was looking like a shadow of himself to turn into Froomebot on next stage.