〜わけない means “no way that…”. わけ means “reason” or “case”. そんなわけない = “no way”, そういうわけじゃない = “that’s not what I mean”, and a rising 〜わけ? asks “so you’re saying…?”.
Is 〜わけない rude? Who can you say it to?
With friends: OK. With coworkers: careful. With a boss or teacher: avoid. わけないだろ is a common comeback; rough with だろ. Casual.
How common is 〜わけない in real speech?
In 1,905 live comedy routines, 〜わけない appears in 572 lines from 153 comedians.