Frozen Factory: Zone 1
Red Star Ring 1: From the starting point, run down to the underside of the area and go straight forward until you see an animal cage in the center of the path, surrounded by spikes. The Red Star Ring is suspended above it. It's high, so you'll need to use the Y button to bounce from the cage while it's still intact to reach it.
Red Star Ring 2: While bouncing across the clouds, you'll see a bell on your left. Chase it a short distance to reveal the Red Star Ring.
Red Star Ring 3: When you reach the area with the large pistons moving up and down, look for three in a row. The Red Star Ring is hanging above the third.
Red Star Ring 4: After you're through the segment with the pistons, you'll be sent through some pipework and be dumped into another area with massive, pink fan blades. Go to the shadow side, and you'll find the Red Star Ring high on a wall to your right. The easiest way to get to it is to find the preceding platform with a jump pad on it and use a laser power-up to fire yourself across the gap.
Red Star Ring 5: There's a boost on the platform directly above the previous Red Star Ring. Use it and follow a linear succession of boosts, jump pads and pipes. You'll eventually need to use Sonic's wall-run to clear a large, floorless gap, at the end of which you'll find the last Red Star Ring.