The cold tap handle on my bathroom sink just kept turning round without turning the water on. Local plumber said that the serations had worn (probably through turning off too tightly). The original tap was a small tap with non-rising spindles. It was replaced with a slightly larger tap with rising spindles i.e. a long neck on the screw/handle. One day later the whole tap was spinning around. Local plumber came back and tightened the plastic nut. He said he couldn't use a metal nut for fear of cracking the sink. He again blamed over-tightening and advised us to switch the tap off more gently saying it would drip for a while but gravity would ensure that any drips would soon stop. This we did but two weeks later the whole tap is spinning around again. Local plumber again blames over-tightening. My point is that we have not tightened this tap any more than the previous tap with non-rising spindles and that lasted 15 years before the serations eventually wore out. Could the tap itself be causing the problem as apparently the rising spindles mean that it is fixed to the sink as part of the mechanism and doesn't have serations or is he trying to explain away poor workmanship?

Why not install a new faucet?

Why not install a new faucet?

That is my final option. Thank you for your help.

From the wording of your post I am guessing you are in the UK or EU. Plumbing is plumbing but some things are done differently in different places.
That said, here in the US we have faucets that function on two principles. The first is the compression type, what you are probably calling a rising stem or spindle.
This type operates like a big screw with a washer on the end. When you turn it one way the screw or spindle goes down and presses the washer down on the valve port (the hole where water comes in). That stops the flow of water. When you turn it the other way, the spindle rises and lifts the washer and the water begins to flow.
There other type is referred to as a cartridge (a non rising stem). The cartridge also has a hole in it. When the hole in the cartridge is aligned with the valve port the water flows. When the cartridge hole is not aligned with the valve port the flow of water is blocked.

The two types of stems or spindles are not interchangeable.

If faucets are built the same way where you are located, you can replace the faucet as Brian suggested or you can replace the plumber.