I've tried to shift them before but failed miserably every time.

I tend to get a new screen put in instead, and i don't go down the insurance route, it might be slightly cheaper (and you might be susrprised here) by paying just the excess but i'd rather just buy my own and with luck have no claims of any sort to report at insurance renewal.

For screens avoid the household names and mass advertisers, instead have a word around and see who the local bodyshop or car trade uses, there will be a company who looks after these places, a chat with them can provide you with a new screen fitted at amazing prices, especially if you have a fairly common car, my last screen on a very uncommon car was £145 IIRC, well worth the money when the one it replaced had 15 years of scratches and stone chips.

Thanks galileo, I am worried about doing more harm than good if I try and polish it out. It was caused by some grit under the wiper, I can't see it at all on the outside, just on the inside, it's on the passenger side but being as I know it's there I keep looking at it. I think I will just try and forget about it.