"Scrubber" is probably of British origin, used as a detrogatory term to describe scullery maids or other low-ranking women in domestic service. These were often young women viewed by higher-ranking people as licentious and dirty (particularly their hands). Due to the lack of occupations for women in the 19th and early 20th century women usually were forced to work in service or as prostitutes, and many had dealings with both. The actual term "scrubber" may refer either to her occupation of scouring floors, or to the actual implement used to clean the floor - the scrubbing brush.
You're a right scubber!
Look at that girl at the end of the bar; she's a scrubber.
Mr. Forthingshire: Rather well dear chap. During this morning's hunt, I was fortunate enough to come across a herd of scrubbers.
Mr. Biggins: Oh, very good. I hope you showed those scrubbers a thing or two.
Mr. Forthingshire: That I did dear boy, that I did. One might say the Western district of town is now scrubberless.
Mr. Biggins: Splended old chap! Well, Marjory is cooking a pot-roast tonight, so I best be off.
Mr. Forthingshire: Toodle-pip!
The word scrubber refers to those of the lower working class, characterized by wearing shell suits (full tracksuits for those who don't know), living on council estates and benefits. Scrubbers are notoriously tight with what money they do have, have no regard for personal hygiene and tend to be loud, rude and arrogant. They are very similar in this respect to chavs, only less clean and with a tendency to have a much higher opinion of themselves. When questioned or confronted about this, they can be defensive and violent. Alcohol and drugs are almost certain to be in the vicinity of a scrubber, the money that regular people would spend on hygiene, food and other normalcies is almost always spent on alcohol and drugs, usually in huge quantities, even in front of their usually many children, which sadly results in those kids growing up knowing nothing better.
Scrubber: Lend me some money mate, I need to clean ma howse and stuff
Random Bloke: No, go away
Scrubber: Oh go on mayte, I need it to clean ma howse and shit
Random Bloke: No? How come you don't have enough to clean your house anyway?
Scrubbers: I spent it boowze and stuff
Random Bloke: Piss off you scrubber