Is there any way to generate useful electricity which is completely pollution-free?
Well, the manufacturing of so-called 'green' electrical generators still creates pollution. So, I don't think there's any 100% pollution-free method. However, when compared to coal or natural gas power plants, wind and solar generators produce next to nothing in terms of pollution. Actually, they don't really 'produce' any pullution (so long as your don't count noise or visual poullution), but again, to make them does produce some pollutants.
hydroelectric is the most clear one i think.
There are a bunch (solar, wind, hydro, etc) that create no pollution, but that doesn't mean they don't have environmental impacts. The only one that is generally ok is solar because most people aren't too concerned about desert ecosystems. All other power plants are generally challenged on environmental grounds.
if by pollution you mean creating an impact on the environment.. no
is there a better way than coal and oil? yes.
wind is a good tool, and can generate the energy we need if deployed more. but the turbines are gaudy and change local weather patterns.
hydro power destroys ecosystems above and below the damn.
solar requires to much surface area.
geo-thermal is a localized energy resource and cannot be deployed everywhere.
hydrogen is an ideal "battery" for energy, but the production is difficult and requires more conventional ways of producing energy, though it can be made from the "green" energy methods.
I would say that today, the best solution is nuclear power. it produces a dense waste product compared to oil and coal. the radiation and toxic elements are very concentrated as opposed to oil and coal and it can be reprocessed to make it safe for transport to storage or reuse.
the ideal nuclear solution would use thorium. the reactors are safer than conventional fissile reactors, they do not produce weaponisable waste. the fuel is abundant enough to take the world 500 - 1000 years into the future with out worry for fuel. and the waste is low grade and short lived (500 years)
the use or nuclear reactors would only be a stop gap until we can get a fusion system that produces enough power to replace the fissile reactors.
Separate names with a comma.