How much electricity is there in the "power switch" button cables?

The power switch provides an active low input to a logic gate within the power supply circuit.

This provides a low current logic low (ground) gate pulse to a BJT transistor biased to an isolated 5V source with the maximum current likely much less than 10mA. This portion of the circuit is also electrically isolated from the rest of the power supply circuit as can be seen in the diagram.

So to answer your question, No there are no dangerous voltages present.

Source: https://en.wikipedia.org/wiki/Power_supply_unit_(computer)

The wires should only have low voltages on them (<= 12V), so they're safe to touch.

However, you should be careful, because if one of the wires is connected to the standby voltage rail, shorting it to the chassis could potentially damage the motherboard.