Be very careful when repairing a SMPS if you don't know (exactly) what you're doing, you may get hurt badly!
That being said, what's the controller chip U1? Don't measure across the transformer windings with a multimeter, it doesn't make any sense, and definitely don't use the oscilloscope in the primary side.
How old is the power supply? The first thing I would check, beside the fuse, would be the three capacitors on the primary side, above the transformer in the second picture. In particular, the one on the supply pin of the controller.