Solution for Supergirl in We Don't Need Another Hero

Opening the vault

Opening the vault is a matter of raw strength, but as Supergirl says the first time she tries to open it, she's a bit out of shape. So she needs to exercise. In order to do that, go to the abandoned factory and pick up the wrecking ball, then use the wrecking ball on Supergirl. Do it twice.

Defeating the robot at the industrial complex

Then again, this is a matter of strength. You need lots of it. Specifically, you need to have opened the vault, exercised twice and stopped the tank. You also need Supergirl to realize that the robot isn't just a statue, so the first thing you must do is try to walk to the gate. As soon as Supergirl is within the robot's reach, he'll attack her. Once that's happened and Supergirl is strong enough, just use strength on the robot.

Getting the autographed photo

Once you've opened the vault, the Bank Teller will go to the mall and stand in line to get an autographed photo of the Imposter Supergirl. If you talk to her as Supergirl, she'll just get angry with you, so you have to change into Linda Danvers. In order to do that, just move a screen to the left to the back alley, where you'd find the dumpster. That's secluded enough, so use the Linda outfit on Supergirl, then return to the mall. Talk to the Bank Teller and you'll get the autographed photo.

Getting the doll

This one is easy. Just give the autographed photo to the little girl and she'll give you the doll.

Finding evidence to enter the industrial complex

Even if Supergirl is more than capable of entering the industrial complex via her muscles alone, she'll need evidence to even try. (She's not Batman, you know.) In order to get the evidence, fly to the university and give the Supergirl doll to the Nerd. He'll tell you that he'll run some tests on the doll. You need to leave the room and return before you can talk to him about it or he'll complain that you must be new at adventure games. Once the Nerd tells you what he found, you can enter the industrial complex (provided you get rid of the robot first of course.)

The maze inside the complex

The maze of corridors isn't a maze per se, you can't get out of there by merely walking. You need to use your senses. Namely, listen to each wall until you hear something. (The right wall changes randomly ever time you enter the room.) Once you've found the right wall, use Supergirl's strength on it.

The Tiny Killer Robot

This isn't so much a puzzle as a joke. Supergirl will refuse to attack the Tiny Killer Robot on account that he's too adorable. You need to talk your way out of this one. And you need the help from the prisoners at the sweatshop. First enter the room with the Tiny Killer Robot, then go back to the prisoners and ask their advice on dealing with the Tiny Killer Robot. They won't help much, but go back to the Tiny Killer Robot and tell him that the prisoners just laughed at you. You can't miss from there.

The elusive Imposter Supergirl

Once you enter the Imposter Supergirl's office, you'll find her there but Supergirl won't be able to catch her. If you use Strength on her, she'll just pop somewhere in the room as soon as Supergirl gets too close. So, again, it's a matter of using your senses. Use Supergirl's X-ray vision on the Imposter and Supergirl will realize that it isn't the real Imposter, but a hologram, so she'll find the hologram projector and destroy it.

Finding the Imposter Supergirl

Well, you got rid of the hologram, but where is the Imposter Supergirl? She's behind the bookshelf. Either listen to the bookshelf or use X-rays to find her, then use Strength to move the bookshelf aside, and sit back for the game's ending.