Voltage ripple on power lines after 6502 CPU is reset
This could be normal for processor, but anyway shows probable deficiencies in the design of the board and circuit. Datasheet states standard 5 V ± 5% guarantee, thus you seem to be formally within spec.

Digital electronic devices, by their design, consume power in discrete pattern, and this is the source of the noise on the power lines. You have 0.1 uF bypass caps, you may also need several electrolytic caps of ~10 mF in several locations on the board, and you have already placed them.

It would much help if you look at the board and analyze power track paths to find most loaded locations, and ensure that conductors are of appropriate width (in case of PCB).

The fact that your retro-PC does not work properly may not be related to power quality. You mention reset, not detailing what it means - after power cycle or reset button press. I must say that reset circuit is inappropriate - it consists of pull-up resistor and button only, I do not see smoothing capacitor and ideally Schmitt trigger. Thus it may simply happen that 6502 can not reset properly.

Here's relatively complex example of the reset circuit for exactly your processor. Here're more circuit diagrams. There're a lot of variations, but the idea is the same - active low (in your case with 6502) must happen steadily within minimum full 6 clock cycles (see datasheet's page 7), and then go inactive without any bouncing, and stay there for normal CPU operation.