I found a list from ps3trophies:
Hey guys, here is a list of the crown requirments for Hot shots golf hope this help some of you work on this long plat.
1-1: win with 2000 points or more
1-2: use a rising shot*
1-3: Hit GIR in 1 Stroke on a Par 4 and Win
1-4: win with par on frequency 100%
1-5: win with over 3 bogeys
1-vs: win with an eagle
2-1: just impact frequency 50% or more
2-2: fairway keep 100%
2-3: add a spin on all your shots
2-4: win with 2000 points or more
2-5: don't use a spin shot
2-vs: win within the first 3 holes
3-1: win with 10m or more putt/par or better
3-2: use a pin shot*
3-3: win with 3 birdies in a row
3-4: don't land in any roughs
3-5: win with 3 birdies in a row
3-vs: all holes must be done within 1 putt
4-1: total score must be -3 or less
4-2: fairway keep must be 80% or more
4-3: par on frequency 100%
4-4: total score must be -4 or less
4-5: win with no OB/WB
4-vs: win with consecutive birdies
5-1: 1m better pin (par on or better)
5-2: don't use spin shots
5-3: par on frequency 100%
5-4: get 1500 points or more
5-5: get a chip in birdie or better
5-vs: no bogeys
6-1: 2m better pin (par on or better)
6-2: no OB/WB
6-3: no bankers
6-4: 3m better pin (par on or better)
6-5: get a birdie or better on the last hole
6-vs: don't use power mode
I've also figured out what the two shot types are:
Rising shot - You must put tremendous top spin onto your shot and your ball has to traverse up the pin.
Pin shot - Your shot has to hit the pin.
Not sure what OB/WB and DP/WB means though.
