What properties would detergents need to have in order to help remove oil from birds?

They have a hydrophilic and hydrophobic end

Explanation:

The structure of detergents enables them to perform their functions as a cleansing agent for removing stains. Detergents are more preferable to soap for cleaning because they are indifferent to water hardness and can retain their cleansing properties in any medium.

A typical detergent has a hydrophilic head (-SO₃⁻Na⁺)  and a hydrophobic tail, a long chain hydrocarbon from C₁₂ to C₁₈ or a benzene ring having a long chain hydrocarbon as its substituent.

This structure enables the hydrophilic head to bind with water and makes the detergent soluble. The other head head attaches to the fabric, in this case the feather.

The property that makes detergents able to remove stains lies in the structural arrangement of their component elements.