Each type of LED gets attached a little differently, and some need a bit of advance prep. Here's how to connect each type to the circuit:
If you look closely at the Circuit Stickers you will see that the rounded side is marked positive (+). The pointy end is negative.
The Circuit Stickers need no preparation. Just peel one of the stickers off the backing paper and press it firmly onto the traces.
Adafruit LED Sequins
The positive and negative ends of the sequins are clearly marked (+ and -). The LEDs come attached to a board. Snap one off.
If you have Z-axis tape, cut a piece roughly the size of the sequin. Peel the paper off one side of the tape, and press the back of the sequin to the adhesive. Then peel the remaining paper off the back.
As you saw if you performed the throwie test, the longer wire on a standard LED is usually the positive lead. The side of the LED bulb near the negative lead also usually has a flat spot.
Bend the legs of the LED out to either side. (A pair of needle nose pliers can help.) Then bend one lead facing front and one facing back. If they are very long, you can bend them in half again.
Place the LED so the positive lead is sitting on the positive trace and the negative lead is sitting on the negative trace. Attach each lead to the trace using additional pieces of conductive tape, pressing firmly.