Stoves and Fuels

Burner on Gas Cartridge
Pros: the Camping Gaz Bleuet or Globetrotter style burner with a pierceable cartridge isn't really an awful lot of use to the backpacker, simply because you can't demount the cartridge until it's completely used up - which makes packing a bit awkward, plus you have the added delight of watching boiling times get longer and longer. The latest versions allow small lightweight burners to connect straight on top of a valved cartridge, so it can all come apart for ease of packing, at the same time allowing you to swap between cartridges so you can use the more depleted one for simmering, the newest for fast boil-ups.
Cons: not usually so brilliant on stability, although the problem has been addressed to an extent with some cartridges being fatter and squatter, thus providing more of a stable base.

Burner on Flexible Pipe
Pros: makes for a much more compact setup where burner and cartridge can be separated when they're packed, and the burner in use is also closer to the ground - usually on fairly wide-spaced legs - so it's a good deal more stable.
Cons: you need to take care to inspect the flexible tube fairly regularly to ensure it and the points where it joins the burner and valve assemblies are secure and not leaking

Burner on Rigid Pipe
Pros: if you're going to have the fuel reservoir separate from the burner, a rigid tube is by far the best means to connect the two. Apart from the fact that it's more robust, it also turns the fuel tank into an outrigger for the burner, providing the best overall stability.
Cons: none.

The kind of fuel you burn comes partly down to local availability, and partly what you feel most happy with. Backpackers of a nervous disposition might prefer to give petrol stoves a wide berth and opt for the ease and convenience of a gas stove. On the other hand, if you take off to wilder parts of the world, you're really governed by what fuel you can actually lay your hands on - for much of the third world, paraffin will be the most readily available. 

Pros: no messing about, and no mess, too. Meths (methylated spirit) is the absolute foolproof fuel, requiring no pressurised burners, valves or anything which might break down or clog up. The best example of a meths burner is still the Swedish-made Trangia, with its integral windshield/pot stand. Meths is volatile, and it's easy to clean up spills.
Cons: the flame is difficult to see, and it tends to be rather smoky, so expect blackened pans. You can reduce the effect by mixing a little water (around 10%) with your meths - makes the fuel go further, too. Meths isn't always so easy to get hold of, and you may even have to sign a poisons register in some places before they let you have it!

Pros: burns in a pressurised stove - not the same calorific value as petrol, but it's a good performer nonetheless. If you're trekking in third world countries, this is likely to be your best option, as this stuff will be available even in places where you might find it difficult to get hold of petrol.
Cons: needs a separate fuel such as meths or alcohol paste to light the stove initially (unless it has a built-in wick). Needs care when lighting, as, like petrol, it can flare up until the fuel flowing through the delivery tube is vaporising properly. Doesn't evaporate readily, and it's rather smelly, so don't spill it!

Petrol (Gasoline)
Pros: stoves capable of burning automotive petrol offer a good fast boil-up, using a fuel which is readily available almost anywhere.
Cons: tends to clog up jets and fuel tubes, so don't use it unless you can strip the stove down for fairly regular cleaning. Low octane fuel is advisable - don't burn lead-free unless your stove is specifically made for it. Lighting in cold weather needs extra care, as it has a tendency to quite dramatic flare-ups until the fuel running through the delivery tube starts to vaporise properly.

White Gas (Coleman Fuel)
Pros: the best fuel for petrol stoves - clean, additive free, and a lot more pleasant to cope with if you have to deal with spills.
Cons: expensive, horrendously so in some places, and that's even if you're lucky to find someone selling it!

Butane / Propane
Pros: quick and easy to light - some stoves even come with built-in piezo ignition to give you all the comforts of home. Used to be regarded as a lazy burning fuel best suited to low level summer conditions, not much use in cold or wind. But recent technological developments have made gas stoves some of the quickest boilers on the market, whilst the addition of a proportion of propane gives the fuel the ability to burn well in the cold.
Cons: boil-ups tend to take longer as fuel and pressure in the cartridge becomes depleted.

If you have the right multi-fuel stove, you can burn an awful lot more besides petrol and paraffin. My staple fuel for many years was white spirit (paint brush cleaner) - a bit smoky on start up, but as good as paraffin when it got going. Others worth a try are diesel and aviation fuel. My favourite for the sheer daftness of it all was a half bottle of vodka (I drank the other half!)