How to check SMD capacitor for being shorted, without disconnecting it from the electric board

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Without a schematic, you will need to engage in some sleuthing to find the ground plane. A good place to start checking would be the negative (-) pins of the large electrolytic capacitors, which are clearly indicated on the plastic sleevings of the capacitors (the lighter-coloured section).

Using the plated-through-hole mounting pad may or may not work, because they are not always connected to ground. You can check this by measuring between the electrolytic capacitor negative pin and the mounting pad.

You can still take direct continuity measurements across the capacitors. A short is a short.

One final point to remember - these capacitors are almost always connected in parallel with other components in the circuit. A short indicates that one or more of the devices on the circuit have failed short - not necessarily the capacitor.

The most common failure mechanism for ceramic capacitors to fail short is mechanical stress causing the ceramic layers to crack and internally short out. Unless you dropped the assembly, I doubt the caps are bad. If they were exposed to excessive electrical or thermal stress, you would see burning, discoloration, etc.

The measurements you took already - all with some resistance or beyond the DMM measurement capability - imply that none of the circuits connected to the capacitors you checked are shorted. Your problem is likely elsewhere, or different in nature (i.e. not a short).

Where I would start is putting your DMM in DC voltage mode, plug the AC power in and check for DC on the logic board by carefully measuring voltages across the ceramic caps and e-caps on the logic board. DO NOT PROBE THE POWER SUPPLY. If you touch the wrong area you could get electrocuted. Also be very careful to not short your probes together when measuring, or you may get a really big spark.