I have no idea what you'd award Trophies for in PlayStation Home, but soon you'll be able to collect them in the social game on PlayStation 3.
The community manager for PlayStation Europe, Chris Owens, confirmed that version 1.86 of PlayStation Home is coming this Wednesday, March 12, and will introduce Trophies, Sony's in-game awards. The addition will go into effect worldwide and will be retroactive. Sony will announce the Trophy list the same day as the update.
Owens had "no news" about PlayStation Home coming to PS4.