I have done some digging and a lot of watching, and I have decided to break this down into three lots.
Firstly, the best nul point entry before the 1975 rule voting change; where receiving nul point was a lot easier. I will admit I didn't put that much effort into this choice, but I'm happy.
It's Italy from 1966 - "Dio, Come Ti Amo" by Domenico Modugno
Secondly I have chosen my favourite nul point entry from 1975-1998 when the voting changed to the current system, but was jury voting only. This also feels quite nicely as the "second stage" of Eurovision.
After much deliberation, it's the Austrian entry from 1991. Thomas Forstner came last with 'Venedig im Regen' which was totally undeserved. I like a lot of entries from 1991, a few were a bit hit and miss, but Austria shouldn't have got nul pointe!
Austria from 1991 - "Venedig im Regen" by Thomas Forstner
So now I move on to my favourite from 1998-2015, with no points a result mostly of televoting. There have only been 4 nul points since 1998, including the UK's very own in 2003. Ah I have such pride! But this is a relatively easy choice. Austria take my favourite again because there is no way they should have got nul point last year. Take it away the Makemakes!
Austria from 2015 - "I Am Yours" by the Makemakes