Murphy's Oil Soap is not good for birds. Especially birds that fly free around the home. If your bird likes to pick toys and food off the floor this can be toxic to them if MOS is used. The oil can also be harmful to their skin. For hardwood floors, you can mix bicarbonate soda or club soda (without quinine), with a touch of all natural almond oil (unscented). This will clean the wood floors, restore shine and give a protective coating. This will NOT give off any scent, and if you have a bird that flies freely and occasionally settles on the floor or what not, the oil will not harm them and or the soda is harmless if they pick a toy or piece of food up from the floor. The soda also works so well it makes it so that you don't have to clean the floor as often. This solution minus the almond oil is also an excellent bird friendly cleaner for the cage. If, after removing the bird from the cage, you lightly mist the bedding with the solution, this will prevent the dander from entering the air when removing bedding. This keeps the allergens down so that your allergies are not antagonized as well as your bird's respiratory system is not compromised.
I have a friend who would take her bird into the shower with her and the bird would perch on the side of the shower stall playing in the water while she showered. This is harmless....AS LONG AS YOU ARE NOT USING SOAP...most soap products will affect your birds skin and feather makeup and this can also be bad for their eyes and the fumes are not good for them. It's best to ask your vet what "soap" to use when cleaning your bird or to use just room temperature to SLIGHTLY warm water.
Peroxide should not be used when cleaning around a bird's environment either. This dries their skin and also can be harmful if ingested at all.