When I was a kid, whenever I'd use me as a subject pronoun (e.g., Me and Erin went to the mall) Dad would say "And did you do that before or after you went to English class?"
In English class back in the day, we learned about logical fallacies. To wit:
An attack on the speaker rather than an attack on the argument. For example:
Opponent: Vaccines cause autism
Me: I have 500 scholarly articles that demonstrate otherwise.
Opponent: You wouldn't understand, you're not a parent.
Argument claiming we know too little about something to say it isn't true.
Opponent: Of course it's possible vaccines cause autism. There's too much we don't know about autism to say otherwise.
Me: Of course children with autism can fly. We don't know enough about autism to say otherwise.
Argument from authority
Claiming that because someone who is smart believes something, so it must be so.
Opponent: Legions of parents know in their hearts vaccines cause autism.
Me: Legions of cranks all know in their hearts they've been abducted and probed by aliens.
Confusing association with causation/ post hoc ergo propter hoc:
Claiming that because one thing is associated with another, one thing causes the other.
Opponent: My kid was perfectly normal until he got vaccinated when he was two.
Me: Your kid also wasn't potty-trained until after he got vaccinated when he was two. Autism symptoms begin to show around the age of two, right around the time when kids get a big round of vaccines. This is also true in kids who don't get vaccinated.
Courtesy of The Skeptics' Guide to the Universe.
Those are just a few. But to those, I would add something new for the age of the Internet:
The tearing apart of someone's argument because they use bad grammar while making them. I've been so guilty of this one.
Opponent: Your wrong.
Me: It's you're wrong. If you don't know the difference between your and you're, then clearly your argument has no merit.
Of course, that's not actually a fallacy, because it's totally true, right?