If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:
- Let $a = b$.
- Then $a^2 = ab$
- $a^2 - b^2 = ab - b^2$
- Factor to $(a-b)(a+b) = b(a-b)$
- Then divide out $(a-b)$ to get $a+b = b$
- Since $a = b$, then $b+b = b$
- Therefore $2b = b$
- Reduce to $2 = 1$
As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.