Uncommon decorations have a blue star shown next to them in their page of the coral section of the Aquapedia. They can be received from the Hermit Crab with both purple and gold tickets.
Rare decorations have a gold star next to them in their page of the coral section of the Aquapedia. They can be obtained from the Hermit Crab with gold tickets.
Limited time decorations are available from your Hermit Crab with golden tickets too. These will only appear for a certain period of time before a different one replaces them.
Note: Storing a decoration will reset it's timer - when you take it out of your inventory you will have to wait for it to be ready again, so choose which decoration you store wisely!