Since the Thatcher sell off legislation there is such a shortage of council houses and flats your chance is zero.
I think this is an invocation of 'RudeBoy's Law'; ie, it's always Thatcher's fault, ultimately.
TJ and Ton are right; unless you have children and are on very low income, or have a debilitating illness/disability, or are considered 'vulnerable', most urban councils will not even look at you. Which is correct, really. Council housing should be for those with the most need, and who are unable to secure adequate housing for themselves.
Sadly, the number of people who do deserve and need social housing is ever increasing. Thatcher's lot may have made a fast buck, from the sale of social housing stock, but the reality now is that the cost of housing the needy in privately owned housing is costing the Nation a shitload more than Council housing ever did.
Might sound a bit harsh, but if you can't afford to buy, then you should either make efforts to have a more financially rewarding career, or just accept it, and continue to rent. Renting is not that bad; you may not end up owning a property (something that does not in any way bother loads of people in France, or the US, for example), but then you don't have the hassle of having to pay for every tiny little cost associated with home ownership. And you can relocate at the drop of a hat, almost. Renting does give you freedom to change your lifestyle/job/career/surroundings whenever you wish.
You're still in a position to put a roof above your own head. That's not so bad, is it?