Lucky Isle is an island that will start appearing once you have beaten Fury Bowser at the end of the main story. The first time Bowser appears in the post-game, you will see a short scene of a small floating island that has five Cat Shines on it. If you open your map, you should notice that there are small circles with a grey exclamation mark on them, which are actually the locations in which the Lucky Isle will appear. This will change to a yellow icon that has a black exclamation, for when the Lucky Isle appears. There is a bit of a catch, though, as the Lucky Isle will only appear whenever Fury Bowser is awake. Combine that with the fact he will go back to sleep upon getting a Shine means that you will need to wait for Fury Bowser on five different occasions to get all of the Shines from Lucky Isle.
Lucky Isle Cat Shine #1
One of the first locations where you can find the Lucky Isle is right next to Pounce Bounce Isle. Take the first red Jump Pad and you should see a lone grey brick block right there. Wait there and when Bowser is out and about, the Lucky Isle will also appear there. Grab a Cat Shine when it appears.
Lucky Isle Cat Shine #2
Another location for the Lucky Isle to appear is right next to Slipskate Slope, so head up to the gate and you can actually just sit there by the main gate. Wait by the gate for Bowser to wake up, then the island will appear right next to it. Grab a Cat Shine when it shows up.
Lucky Isle Cat Shine #3
Travel to Crisp Climb Castle and grab one of the Propeller Blocks at the base of it. Use this to fly to the top of the first tower, where you have the Flopters bothering you. Make sure you keep the Propeller Block, as you will need it to fly to the Lucky Isle when it appears nearby. Grab one of the Cat Shines when you are on the island.
Lucky Isle Cat Shine #4
This Lucky Isle location is a bit tricky to get to, since it’s located just west of where Fury Bowser makes his “home.” You will need to get on top of the wall in the upper section, which is connected to Pipe Path Tower. So, travel to that location and grab one of the Propeller Blocks, then make your way through the course until you can float down to that large wall. Remember that this is the same wall that has multiple Fury Blocks throughout it. Make your way to the end of the wall (there’s a Shine here) and wait for the Lucky Isle to appear, then fly/float over to it and grab a Cat Shine. Note that you can also use Plessie to get to this spot, where a ramp will appear nearby that can launch you to the Lucky Isle.
Lucky Isle Cat Shine #5
Head to Roiling Rolling Isle and make your way to the first stable platform with a Baddie Box. Wait for big bad Bowser to wake up and the Lucky Isle will spawn right at the side here, so hop on it and grab a Cat Shine.