The term "classes" in fire safety does not refer to specific types of extinguisher. Instead it refers to the material that is burning in the fire. Of the five main types of extinguisher, four are suitable for tackling wood and paper fires.
For safety purposes, fires in the UK are broken into five classes based on the materials involved. The classes are A for solid materials such as paper or wood, B for flammable liquids, C for flammable gases, D for combustible metals and F for cooking fat and oil. Class E, which covered electrical appliances that were still plugged in to the mains, has now been discontinued: a fire involving electrical items is classified by whatever other materials may be involved in the fire. Note that, unlike places such as the US and Canada, the UK does not use particular shapes or colours to indicate classes of fire, instead solely using the letters.
Fire extinguishers in the UK fall into five categories based on the material they contain. Each is indicated by a band of colour that appears on the red background of a fire extinguisher. The five categories are water (colour coded red), powder (colour coded blue), foam (colour coded cream), carbon dioxide (colour coded black) and wet chemical (colour coded yellow.) Water extinguishers may have a dark red band on a light red background, or simply have no band marking. Each type of extinguisher is suitable for a particular class or classes of fire.
Suitability for paper and wood
Water, dry powder and foam extinguishers are all suitable for class A fires involving solid materials such as paper and wood. Carbon dioxide extinguishers are not suitable for class A fires as they do not cool materials. Wet chemical extinguishers are specially designed only for class F (cooking oil or fat) and thus are not suitable for class A fires.
The three types of extinguisher that are suitable for class A fires have differing suitability for other classes of fire. Water is only suitable for class A fires: if used on other types of fire it could create additional risks. Foam is also suitable for class B (flammable liquid) fires. Dry powder is also suitable for both class B and C (flammable liquid and flammable gas) fires and thus is usually the best general purpose extinguisher to have on hand.
Note that none of the extinguishers suitable for class A fires are suitable for class F (cooking oil and fat) fires, which require a specialist wet chemical extinguisher. If a fire involves live electrical items then carbon dioxide is the safest type of extinguisher regardless of the other materials in the fire. Never use water extinguishers if live electricity is involved.