Unless you're among the few with diesel or gas heating, you're going to be burning wood, smokeles coal, or both. So, which is best?
Wood or smokeless coal?
Burning wood generates a small fraction of the carbon emissions that come from smokeless coal, although wood smoke is worse for the lungs of passers-by (something to do with particulate sizes) and it tends to hang in the air on calm days.
Wood doesn't 'bank' (stay hot) for long so is unlikely to keep you warm all through the night, and being bulkier it's also harder to store and keep dry, but then it's usually quite a bit cheaper than coal and it smells better when burning.
You might want to experiment with using wood and coal according to need - try wood during the day and smokeless coal overnight, for example. First, though, read on...
Built-up areas and the Clean Air Act
Boats on the waterways are exempt from the provisions in the Clean Air Act that ban the burning of house coal or wood in smoke control areas, provided that the smoke from your chimney is not persistently black. That said, if your chimney smokes heavily in any shade of grey then the local council's antisocial behaviour team or pollution team can still deem you a 'nuisance' and take action against you.
Whatever the law has to say, burning anything other than smokeless coal in built-up areas tends to bother neighbours, so it's best not to burn wood (apart from kindling) when moored up where houses are right next to the canal. In London, for example, some local residents have complained repeatedly about wood smoke.
And burning house coal and rubbish?
House coal furs up everything - the chimney flue, your lungs, and the patience of any passers-by. It also has no advantages over smokeless coal. In any case, the fuel boats tend not to stock house coal, so it's best just to give up on the idea of using it..
Don't burn rubbish. Anything that gives off black smoke stinks and is illegal - you'll get done for burning it.