Dell laptop not switching on

Before this started, did you notice if the battery was charging? If it wasn't, I would suspect that it has something to do with a bad AC adapter. If that appears to be working, it may be an issue with the internal PSU.

...notebooks don't have internal PSUs.

I think he means the internal charging circuit.

Disconnect mains , remove battery and press and hold the power button, sometimes works by forcing the chraging circuit to reset, put in battery, then mains and pray.

Failing that, if under a year, contact Dell, they have had a bad batch of fubar batteries of late.