The laundry room appears for the first time in the first episode of Season 2, The Bad Fish Paradigm.
Usually random conversations that advance the plot occur while the characters clean their clothes. In the "The Panty Piñata Polarization", Penny and Sheldon conducted an ever-escalating war against each other with Penny using all the washing machines to disrupt Sheldon's laundry schedule. In "The Dead Hooker Juxtaposition" Penny confronts Alicia in the laundry room about her "using" the guys. In "The Psychic Vortex", Leonard tries to get Penny to read a book debunking psychics which she refuses to do so he tosses it into the trash can.
It normally has Sheldon and Penny discussing topics like Sheldon's need for Penny to do or not do something. In "The Large Hadron Collision", Sheldon shows Penny a Power Point presentation with reasons why he should go to Switzerland with Leonard instead of her. Sheldon also hugs her for one of only two times (three times by Sept. 2013) after she tells him that he can go the Switzerland with Leonard. Penny also finds Sheldon doing Howard's laundry in "The Hawking Excitation", commenting on his leopard striped man panties. Leonard also runs into and argues with Penny in the laundry room.
In "The Cooper Extraction", Penny is shown trying to seduce Sheldon in the laundry room during the alternate reality.