My dream is to have a horse win the Belmont. I don't give a rat's rear end about the nuthouse they call the KY Derby. The Belmont is the TRUE test of a great horse in my eyes. The Travers also means a lot to me. However, I highly doubt everyone here shares that opinion...
Like you mentioned, the Epsom Derby is probably thought to be a much higher honor for someone raised in Europe.
Depends on where you are in the world, I suppose. I'd be happy enough to win the Prix de l'Arc de Triomphe or the Irish Derby or the English Derby or any of the Triple Crown races. For racing over timber, get Marylanders in the room with Virginians and discuss Gold Cup vs Hunt Cup.
In America, the Derby is most coveted. It's the race that the general public always asks any thoroughbred racing person about -- "Did you have a horse win the Kentucky Derby?" Most Americans don't know other races. People who have connections to a Kentucky Derby winner take a leap in public recognition.
For racing enthusiasts - it could be so many incredible races in Europe, US or Middle East. Unless I am leaving some place out.
As pointed out, it's going to vary quite a bit geographically and also you mixed jumping and flat racing, which have have their own "coveted" races independent of each other.
The Kentucky Derby is obviously the most coveted race in the US, witnessed by the fact that owners go all out just to have a horse in the race (often when they arguably don't belong) and it is normally oversubscribed.
Personally I would rather win the BC Classic, as it it WFA against the best dirt runners in the world over the classic 10f.
The Arc is the most coveted flat race in Europe, and arguably the world.
The Derby at Epsom would rank high among British and Irish people, but it doesn't suffer from the same mania of having to have a runner in it that the Kentucky Derby does. A Derby winner once he has the Derby under his belt, has to come out and do the business against older horses and cap his season off with the Arc.
For Stayers, the Gold Cup (2m4f) at Royal Ascot is the blue ribband.
For jump racing, the Cheltenham Gold Cup is the most coveted race for chasers, at least among the British and Irish. It is the best of the best going over the classic distance (3m2f) at level weights.
The French would say the Grand Steeplechase de Paris.
The Grand National is probably the most famous race in the world. However, it is a handicap with a 25lb weight spread, over unusual fences at an unusual distance, and rarely do the best chasers take part in it and is often won by a horse that most people have never heard of before and hear of again.
Winning the Gold Cup you can claim you have possibly the best chaser in the world. Winning the Grand National often means you had a lucky plodder that got in at a low weight.
The Champion Hurdle at Cheltenham is the most coveted prize in Hurdling.
The most coveted race in Australia is probably still the Melbourne Cup (though as a handicap it suffers from some of the same problems as the Grand National).
The Golden Slipper, 6f race for 2yos at $3m, seems to be the race a lot of owners in Aus want to win these days. Horses are sometimes retired on the back of just winning that race.
Australia is an odd case in that their most storied race, and richest at $6.5m, is at an unusual distance (for Australian racing at least) of 2 miles. Outside of the big Cup, the Australians are more obsessed with speed and sprinting. The fact the in the last number years the Cup has been dominated by either European owned or bred horses (with some Japanese thrown in) has caused a bit of crisis of conscience in Ozzie racing circles.
In Japan, the Japan Cup (12f), the Arima Kinen (12.5f) and Takarazuka Kinen(11f) would be the "big three".
In South America, the Gran Premio Carlos Pellegrini (12f on turf) run at San Isidro in Buenos Aires would be to South America what the Arc is to Europe.
I think probably as an individual race, the Kentucky Derby is one that you would probably get the most trainers/owners overall of flat racing (in the English-speaking world at least) saying they'd really want on their resume. EVERYONE knows it. Heck, given what Australia and Japan are willing to pay for Derby winners when US studs don't want them they might appreciate it more.
I think the TITLE for a horse most American trainers and owners would say they covet the most is Triple Crown Winner. Any single one of the three is prestigious. Winning the BC Classic or Turf is nice. Winning the big three? Only, what, eleven horses have done it.