Only twice. The first (because I happened to be in Adelaide at the time) was the final stage of the TdU in 2009 - the Lance comeback edition. The other was the 2nd or 3rd last stage of the Giro in 2015, standing on the side of the road in the village of Fénis.
Only been at the small pro races (Europe Tour or smaller) as a comissaire (referee) or at the race traffic control or a head of the whole race traffic control. Quite intense things to do, personally feel super tired after these sessions. There's not many seconds time to 'watch the race' in those jobs, really. Never been at the big races. My cranked view is that best vibes from the race you get when sitting in a pack, but for dat you must do dis and dat, else than that it's Eurosport. Well Worlds maybe if weather is nice and good beer, milieu..
Troughout the years I've been to several stages at La Vuelta, mainly at the finishing stage. I recall watching the finishing ITT as a 10 year old the year Casero won la Vuelta and then some sprint finishes.
I also went to the start of the first road stage some years ago. Funny that Tom Dumoulin was sitting on his bike outside the bus and no one knew him but myself. I was able to take a photo with him. 21 days later he had almost won the race. And I remember the surprise on the face of Maxime Monfort when I said "Allez Maxime" while he was heading to the start. He even turned around to see who I was
I have seen two TdF stages and lots of stages from Tour of Denmark. The first TdF stage was the TTT in 2009 around in Montpellier which Astana of course won (Contador, Armstrong, Klöden, Leipheimer etc) where Armstrong missed the yellow jersey with under a second if I remember correctly. It was an insanely hot day as well and I was standing in a 180 turn where multiple riders crashed, so we got a lot of action that day, lol. Good times
The other TdF stage was in 2011 on a medium mountain stage in Italy before the big Alps on the Pra Martino, the last climb of the day to Pinerolo. Eddy the Boss won that day and Voeckler went into a garage. I think it was that day that the Schlecks complained a lot about safety and all that.
In 2018 I was at the first stage of Tour of Denmark where I met and talked with Wout van Aert. We took some pictures and talked for a good 5 minutes and I remember I said to him he would win Roubaix next year haha. If im not mistaken he was kinda close. But nobody was really interested him in that time as he wasnt that well known for the casuals danish fans and he rode for an obscure Belgian team, Crelan. He won that stage and the overall, so if I were to meet him the next day I probably wouldnt be allowed to talk to him right before the race started like I was that day.