I recently took a trip with Garrus to the top of the presidium where we spent some time seeing who was the galaxy's best marksman. Being naturally competitive, I took the last shot rather than missing on purpose. Now I'm wondering, did I miss an opportunity to unlock Garrus' second squad power here? It would be nice to know when all the unlock conversations take place.
It seems most of the powers are unlocked from conversations you can have from the "investigate" option with them on the Normandy. As long as the squadmate is alive you should be able to talk with them, though it seems you can only unlock some after certain events.
From my experience you just talk to each squadmate and take every conversation option and they should give you one of them. You might have to use the squadmate more to get the bonus powers, especially the second one. I got Garrus' power and I take him all the time.
Additionally it seems you get their second power after they invite themselves to your cabin. They'll mention having a chat in your quarters at some point and you can use the intercom in your cabin to invite them in. After a cutscene plays you'll get the power.
I managed to get Tali's power even though I hadn't yet used her as a squad mate. I got the invitation immediately after she came aboard the ship at a certain, Quarian related part of the plot.