How to avoid shorting the boards while using Probe or Multimeter?
Use a microprobe. The smaller the width of the probe. The less likely you are to hit multiple pads from its radius alone
A sharp one is possible. If it can stab into the solder. It means its less likely to slip. In the past I have used sharp single point pogo pins for this end
Set up the measurement so you do not need to look away from where your probing. The moment it leaves your vision your hand positioning gets much worse. If you can hold a probe to a 0.1mm spot when seen. Your probably closer to 2mm unseen if you cannot stab the board for some feedback.
Use something similar to varnish or heatshrink to make sure only the very end of the probe is conductive. This will reduce the sides from touching other parts of the circuit.
In complex boards. While the power is off tack on some measurement wires. I prefer rigid ones like resistor legs. But for 01005 and 0201 size it would be more likely to be enamelled magnet wire to reduce the risk of tearing stuff off the board. Then you measure those wires.


And finally if you get any say in how the devices are made. Yell at the designers to include some uncoated vias for measuring the power rails with a small exclusion area of say 0.5mm. Makes things much easier for barely any change to layout. And as the probe can rest in the hole your less likely to drift.