Boris, the power supply is a BESTEC, one of the worst ever. HP and eMachines used them, and some dolt in Dell's purchasing department made the decision to buy them for use in Inspiron 530s and other machines.

Bestec power supplies have a terrible feature. If a Bestec power supply fails, it sometimes takes the motherboard out along with itself. If you are lucky, the 530 motherboard will still be good.

First step, since you do not want to retrain your elderly mother-in-law, is to replace the power supply. The form factor is generic, so any decent power supply with a 24-pin power connector and 4 SATA connectors will do. DO NOT BUY ANOTHER BESTEC.

If the power supply replacement does not do it, contact me at . I have one or two 530 motherboards that I have pulled from 530 systems taken out of service. Will sell it to you at a modest price + exact shipping.

On Wednesday, October 18, 2017 at 2:29:37 PM UTC-4, Boris wrote:
Hi,

Machine is a Dell Inspiron 530 desktop, circa 2012, running VistaHP x86,
belonging to my 93 year-old mother-in-law. She called me to day and said
"It's broken".

The power button does nothing. When I hold in the power button, there's no
sound whatsoever. The fan in the back of the power supply does not spin,
but the little green light on the back of the power supply does glow green,
When unplugged, the light goes out. Other than that, nothing makes a
sound, and nothing else occurs.

Here's the power supply:

https://postimg.org/image/4hdxxqbwkr/

Sound like a PS issue?

TIA