This might depend on exactly what you consider to be a 'local' Council, district, county, unitary and whether or not that council has contracted out any/all services and what discretionary services they provide. But for starters how about (and I am aware more 5):

Refuse collection
Refuse disposal
Street scene - litter picking etc
Building maintenance
Highway maintenance
Vehicle maintenance
Ground maintenance - landscaping-arboreal/tree maintenance
Sewage plant maintenance - confined spaces
Events

The world is your oyster!

Whats the background to your question - more info needed -do U mean safety issues the council faces internally or coming from external sources to the council


none-competent council people providing advice to others inside the council - this is rife in some councils

*children / special needs and their problems
*violence to staff - this is also rife in some areas especially in schools and social services
*general public making claims/ attitude of GP
*schools that are going it alone
*accusations against council staff
*current cuts in finance and jobs

to name but a few

In my last job, one of the teams I looked after carried out needle sweeps in both void and occupied properties. This is risky enough, but once or twice an occupier who was believed to be away to the jail returned unexpectedly while the team were in his house.

I have never seen a group of workers so on the ball with assessing dynamic risks than these (low-paid) cleaners - genuinely inspirational they were.