We had a conservatory refurbished and the window sills were replaced
with plastic coated MDF. The cut ends had plastic moulding glued over
them to cover the bare MDF.

Needless to say, the silicone sealant that was used to stick them turns
out to be non-sticky as far as plastic is concerned and the covers fall
off as soon as you look at them.

What would you suggest to fasten them instead ? The plastic looks and
feels as though it might be ABS, but doesn't carry any markings.

One obvious option would be to screw them on, but it would be neater to
glue them if there is a way that would keep them stuck on.