It doesn't go in the Trophy Case but you can put it straight into your storage cabinet, or in a bag with other things and then put the bag in your cabinet
After you get the heart, if it's with you (in one of your bags), you can re-visit the place where you obtained it and "remember"...it will show the final ghost for you again...kinda neat!
I'd like it if the new housing had trophy type cases, and we could put some of our most beloved objects on display, around our house. Maybe these cases would be protected from people picking the items up, kind of like the icons are.
Mine is also in my cabinet. I too wish I had a display case to keep my treasures in. During the Q&A I got the idea that we may be able to have something like that in the new houses. I hope so.
I miss seeing those notes where the ghosties are found, those and Eyeballery are the only way I'd ever have completed it. Tough quest, but I would love to see more 'traveling' quests like it and the 11 strands of DNA one.
If you have the heart in your inventory, you can see the ghost on any street where there IS a ghost (not just on the 7 streets where you found ghosts during the quest). You have to click on the heart and select "Remember" -- then the ghost will be visible for a brief time.