You could be injecting noise simply by holding the multimeter wires and probes, you may be picking up
some RF or other nearby mains noise.
The oscilloscope probe is in a grounded coax sleeve so it doesn't pick up noise.
Also you could be picking up noise from the wiring you use to sense the supply voltage.
The 1V on powering down is perfectly normal - CMOS circuits below their minimum voltage consist
of a bunch of FETs that are fully off, so not current flows and the decoupling capacitors / power supply
cannot drain any further unless there are other routes to ground.