At the moment I have 12 rare mounts on my main collection (with 113 mounts total), and 1 rare mount on my hunter (with 29 mounts total), and here are their stories:
#109 Black Drake: The black drake is so rare on my server, I spent literally months trying to find a guild or group of friends to take me to get the mount. I could have organized my own or went on another pug (if there was one), but I wanted the mount to be guaranteed. Well, I ended up getting a group of friends to take me... we zerged the boss, killing him with only 8 people. I paid 6000g for the mount and they threw in all of the boss loot for me.
#112 Blue Drake: It took me months to get this.. 5% drop rate my butt! After 50 or 60 something random occs, I finally managed to snag this beauty on my warrior.
#71 Brewfest Kodo: I ran brewfest with a bunch of friends, trying to get the mount. I had seen 2 kodos drop, but no rams. On the second day, I bribed my boyfriend into going on his dad's account (his was expired) to come with me and roll on the mount if it drops. Well, the kodo dropped on the last summon, and he won it and traded it over to me.
#82 Green Proto Drake: It took me around 35 eggs to get this beautiful mount, which isn't so bad. But man, am I sick of aged yolk!
#101 Red Dragonhawk: Well, it's not a rare drop mount, but I only know two other people on my realm with it. I was playing for just under 5 years when I finally got him, and it was around 2 years of serious collecting to get to that point. I never though I would even get my albino drake, let alone my dragonhawk! The gorgeous mount was totally worth all the time, gold and effort.
#98 Red AQ Tank: I just got in a raid group and solo'd the trash at the beginning of AQ... over and over and over. Something like 70 other colours dropped before this guy did. It took me around 260ish kills over the course of 2 weeks to get him. I wish the AQ bugs were usable outside of AQ!
#91 Rivendare's Deathcharger: Took me 78 kills to finally get the mount o' doom. I was so glad that I didn't have to go back.. until now, with my hunter!
#67 Sea Turtle: The sea turtle was my first rare mount. I had under 200 fishing and I was bored of leveling my fishing in Dalaran. So I headed out in search of pools (really difficult since I didn't have track fish yet), in hopes of getting some cool stuff other than coins. I jokingly said to my boyfriend 'I'm gunna go fish up the sea turtle'. Less than 5 minutes and 8 casts later, the achievement pops up and everyone's jaw is hanging open in shock!
#73 Brewfest Ram: It was the last day of Brewfest, and I had killed Direbrew around 70ish times. I ran with only one friend this time, and on the last summon, the ram dropped! He won it and passed it over without even thinking. I tried to offer him gold but he refused, so I went out and bought him a 500g epic gem that he needed. ^_^
#111 White Hawkstrider: It took me 67 runs of MgT to finally see the white hawkstrider drop. Not much of an amazing story here, just lots of relief that I never have to do this place again!
#107 Twilight Drake: I managed to get into a OS25 3D pug, hoping that if we did it successfully that I could get lucky and win the roll... which I didn't. However, the winner whispered me and said he saw me advertising for the twilight and black drakes and asked if I wanted to buy it. I offered him 7500g (sparing only 100g for myself), and my favourite drake in game was mine!
#72 White Polar Bear: On my warrior it took me around 90 days to get... so many snowballs. But on my huntress, I got it after three little days!! Woot woot!
Fiery Warhorse: Back when I wasn't even 80 yet I went to do Kara for the first time. The horse dropped, and I proceeded to have a heart attack for the next several minutes. I rolled very crappily, but my boyfriend rolled 98 and told them to give it to me (this was before you could trade BoP to group members). Everyone said congrats to me, but then a late roller... he got 100. QQ.
I'm still trying for Ravenlord (around 85 runs so far), the ZG mounts (20ish runs maybe?), and the fiery warhorse (1 specific mount run!). Feel free to comment on my stories, or post your own! I would love to hear them!
He must have been tired of that place, too. 
