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Serial numbers are typically on the underside of the bottom bracket (where the cranks go through the frame), on the left rear dropout (where the rear hub axle is attached to the frame) or sometimes on the bottom of the head tube (where fork goes through frame). Serial number is typically useful only for registering a bike in case of theft or for recalls.
Look on outside of left rear dropout (where the wheel slides in) or under the bottom bracket (crank housing). Not very useful or important - your Murray is not collectible at all, parts do not depend on model year, and hard to get any info from serial number on a department store bike. How to determine the year value etc of your bicycle
The semi-circular spacer is what holds the derailleur in the dropout. Take the part that looks like a "C" and line it up on the drop out, the spacer goes on the other side of the dropout. There should be a screw that fastens to the spacer and clamps the derailleur to the drop out.
Serial Numbers are usually in one of two locations.
Turn the bike upside down and look at what is the very bottom of the frame where the pedals and cranks pass through. This is the bottom bracket shell. Serial numbers are commonly stamped here and will be a long series of numbers and letters
Occasionally they are also stamped inside the rear dropout on the left or right. The rear dropout is the part of the frame where the rear wheel is connected.
Most serial numbers are located on the bottom of the bike known as the bottom bracket. Too view the serial number flip the bike over and look between the crank arms on the bottom bracket. If the serial number is not visible it may be under a the guide for the cables and that will need to be loosened. Other places for a serial number, but much less common are the underneath of the rear stays or the rear dropouts.