Means the opposite of "broad shouldered" (to act strong and confident and carry alot of burden/responsility in your stride). People (usually a manager) acquire this title by dodging work and/or making decisions and generally pass the buck onto a subordinate. This is done to avoid blame or to avoid doing a task that may be boring.
Officeworker 1: Dave the manager has just given me a pile of accounts to work on thats going to be externally audited!!

Officeworker 2: But he's more qualified than you to do it! Jeez that guy has got such slopey shoulders!
by The Realist! February 02, 2009
Somebody with slopey shoulders is somebody who deflects responsibility or tasks.

It's a step beyond lazy where the person accused is actively trying to push work onto somebody else.
He wants her to do his work for him? Old Slopey shoulders strikes again.
by AnotherJames November 12, 2008

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