I measure the outlets in my room.
They mostly stay at 60hz (sometimes jump to 80hz in a quarter of a second).
When I start my computer, it goes from 350hz to 450hz like insane
The computer is plugged to a powerbar with everything well grounded. And it only does that when I start the computer, all the other things are not disturbing it.
Also, I have a brand new powersupply...
There's no way you're altering the frequency of your AC household current. As Nigel said, you're seeing interference from another source, most likely a SMPS in your computer, monitor or other device. You could be seeing a beat frequency between 60Hz and the SMPS frequency (or a harmonic or sub-harmonic).
Grid frequency is set by multi-ton alternator rotor's revolution rate.
If you have a loose connection or enough resistance in the connection line the startup surge of the computer power supply will create multiple voltage dips that may get picked up by a frequency counter as additional transitions.
U.S. grid spec limits are -0.7 Hz to +0.5 Hz around 60 Hz but almost always is much much better.
A computer switching supply can have a high startup surge, 30 to 50 amps for a msec of time. If the line voltage slumps too much the switching supply can bounce several times before it finally settles out. Power supplies with power factor precondtioner have much lower startup surge.
Put a heavy load, like a hair drier on max heat, and check how much line voltage drops.
Yeah I have a frequency counter mode on my DMM. Well one night I was measuring the frequency of the secondary of a 120v-9v step down transformer, for making a mains based 1hz timebase. So when I went to measure the frequency, it said 800 something hz! I was so baffled, and BTW I found out the mis-reading was coming from a electronic 12volt halogen light power supply.