Hi - My washing machine drains into a pipe in the basement floor. A utility sink drains into the same pipe a few inches "down stream" from the washing machine. When the washing machine drains, the utility sink backs up with a few inches of water, and then slowly drains, leaving behind dirt and debris. Do I simply need to snake the main drain pipe? Where should I do it from?

Also, what are the chances there could be sewerage backing up into the sink? There is no noticeable odor.

Thank you in advance!

" Do I simply need to snake the main drain pipe? Where should I do it from?"

You need to snake from the roof vent that services the washer.

"Also, what are the chances there could be sewerage backing up into the sink?"

When ANYTHING backs up out of a sewer drain line it Is indeed sewerage that backs up.
Regards, Tom