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Find out what the top winding leads to. It could be that something else shorted out, and that shorted the secondary winding, causing it to burn out. A new transformer may do the same thing unless you check all the secondary devices. If the burnt winding is the primary (110V Red-Red), then more likely the transformer is at fault.
It is possible that all 3 voltages come from the same secondary winding, though this is not common. Check the resistance between the different pairs of windings. If they conduct to each other, then any of the 3 secondary could have shorted the winding out, meaning you have to test all 3 secondary circuits for component failure.
If the windings are all insulated from each other, then there may be windings wound one on top of the other, making it harder to determine which winding was actually burning up.
Regardless, since it was a secondary winding that burned up, you must check the circuits that are using those voltages for failed components. If you can't find anything, you can test each circuit with a similar voltage and a fuse (like 1A or 2A) in series. If the fuse blows with no load on, then that circuit is likely faulty. Based on the size of the transformer, I doubt it supplies more than an amp or two (maybe 3A) for each secondary.
If you use a 1A fuse and it does not blow, check the secondary (regulated) voltages. Putting a load on may then show which circuit is faulty..