I have a Fiat Punt0 1.2 which is almost 18 years old and has 68,000 miles on the clock. I have noticed in the last week that when I pull up at traffic lights, a junction or am in a traffic queue which means the car is stationary that the noise of the engine after a while seems like it might cut out. I have then put my foot on the accelerator to make sure the car does not cut out (the car has not cut out yet but my driving intuition tells me something is up). I wonder what might or is happening to my car.

That could be caused by a series of factors.
To begin with, I assume your spark plugs, cables and coils are all functional.
The culprit could be as simple as dirty air filter.
Also, it can be dirty throttle body.
If that are all clean and working, check the post no.36 in this topic:
https://www.fiatforum.com/punto-ii/...v-2004-rough-idle-mostly-electric-load-3.html
(and also check the entire topic, maybe you can find something that help.
Also, you should check the cooler systems air pressure release valve is working. It is located on the radiator cap, and on pressure higher than 1.2bar it should open. So if you blow air onto it, it should open and let the air pass through. Too much pressure in the cooling system is not good and is causing the engine to stall, the rev on idle even drops to 500RPM.
Also, dirty injectors can cause the issue, post no.16 on this topic may help if you want to clean them:
https://www.fiatforum.com/punto-ii/485772-map-sensor-known-good-2.html?p=4583205
Can you do an obd test to see if any errors are recorded? That would be a good way to start and you should check the stat of o2 sensors with the diagnosis device.