Diesel engines were originally designed to run on peanut oil, but the designer was approached by the oil companies and changed the design to run on mineral oil. Most basic diesels will run happily on vegetable based oils provided the viscosity is right. ie its runny enough.
The government don't want us to use veg oil as it circumvents thier fuel tax. You can register to use veg oil but you have to keep records of how much you use and pay the tax.
Bio diesel is veg oil that has been commercially processed or recycled and sold on as fuel for diesel vehicles.
It does make sense to reuse this oil but the government are mainly interested in TAX !!!
LPG runs cleaner than petrol or diesel therefor being slightly less damaging to the environment, although you are still using a fossil fuel. There are grants available to convert your car to run on LPG but the grant is limited to new cars, not older cars. Dual fuel cars are now available from the manufacturer so they don't need to be converted.
Again the basic petrol engine will run happily on LPG with little modification, it's only when you introduce engine management systems and electronic control of the engine you start complicating the issue.
Other propulsion systems are around but the oil companies don't really want us to exploit them. Ford has shut down it's electric car production the "Think" car is not going to be released for sale.
A small electric smart car would be ideal for single person commuting in Edinburgh for most people. The school run would obviously require a large diesel 4x4 though!