Mucus is a noun referring to the viscous, slippery substance secreted as a protective lubricant coating cells and glands of the mucous membranes. Mucous is an adjective meaning (1) containing, producing, or secreting mucus; or (2) relating to, consisting of, or resembling mucus.
In the phrase mucous membrane, mucous is the correct spelling because it is an adjective modifying membrane.
The tableware, the color of mucus and as bendable as a pocket watch in a Salvador Dali painting. [Fox News]
Between 0.5 and 5 mm, they live off the skin, blood and mucus of the host salmon. [Mayo News]
The warm water felt good against his cold mucous skin. [Village]
The syndrome is a rare, life-threatening disorder in which the skin and mucous membranes react severely to a medication or infection. [Minneapolis Star Tribune]