The op knows it is council land so a search, unless he is wrong, wouldn't be of much help and cost a tenner to boot.
What you have to ask yourself is .. What local amenity would be lost? and if it was what would the impact be.
What could you do with the land once in your ownership? If it is to build an extention then a hefty price might be put on it.
If it is roadside then they could well let you buy/have it but could put a no building clause on it and that could include no fences.
In other words they could let you have it but make you keep it as it is. All you would be gaining is the upkeep of it.