Miagani Island: Trophies
There's a little shelter under the bridge. Check here to find the trophy in plain sight.
On the other side of this end of the bridge is a small, breakable part of the wall. Shoot through with the Batmobile to find a Riddler puzzle inside. You need to get the Riddler bot onto the pressure pad, so start by using the Remote Electrical Charge to shoot the generator and move the destructive wall forward. Use the Voice Synthesizer to order the bot close to the wall, then use the Charge to send it back, destroying the bot. Hit the Riddler switch on the wall to restore it - now on the other side of the wall. Order it onto the pressure pad to release the trophy.
From the bridge proper, you can shoot out a destructible wall with the Batmobile. Do so, then grapple into the alcove to grab another trophy.
Lady of Gotham
Head to statue in the center of the three islands for this one. Use Detective Vision to find a damaged piece of scaffolding, then use Explosive Gel or a glide kick to bust through. The trophy is down the stairs on the other side.
This is on a rooftop in west Kingston, but getting it is a little tricky. Just southwest of the trophy is a parking garage. Drive to the top floor and hit the ramp to get onto a nearby rooftop. From here, you can connect the Batmobile's power winch and rev the engine to free the trophy.