It's not easy to follow which council you should speak to when you have a problem, so I'll attempt a rough breakdown of duties. If in doubt though, start with Blaby District Council and they will either sort it out or point you in the right direction.

Contact for most things, including;
Benefits (Council tax and Housing)
Conservation
Business Rates
Council Housing repairs
Council Tax
Election Information and registration
Licencing (taxi, pub, entertainment, food etc)
Housing needs
Planning Applications and advice
Recycling scheme
Sports and Leisure
Waste Collection
Something else? - www.blaby.gov.uk
More specialised on Countywide services, including;
Education
Libraries
Museums
Rights of Way
Road repairs and highways
Social Services
Tourism
Trading Standards
Something else? - www.leics.gov.uk
Local services, such as:
Leisure facilities
some grass cutting
some litter picking
local concerns?
Something else? - www.countesthorpeparishcouncil.co.uk