To keep myself from adding more and more edits, I'll give some 'special mentions' in form of comments. As this is an extension of the post, the rules stated above apply here as well.

Proper use of logic

This is going to be extended upon below in various fallacies sections, but I feel that it should be stated separately regardless. When conducting a reasoning, it should be direct and without 'cheating'. For example, let's say that I'll set the following logical sentences:

  p - "I am the richest man in the universe"
  q - true

can be easily made into a tautology (logical sentence/formula that is always true):

  p OR q <=> p OR true <=> true

does not mean that I'm actually the richest man in the universe. Sadly.

Redundancy

Once working within set paradigm, it is important to remember about your axioms and their purpose. For example, if I were to construct a following set of axioms:

1. "To draw a straight line from any point to any point."

2. "To produce [extend] a finite straight line continuously in a straight line."

3. "To describe a circle with any centre and distance [radius]."

4. "That all right angles are equal to one another."

5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

(these five were taken from Wikipedia article on Euclidean geometry, link in main post)

6. "God exists." (that's my straw-man addition to present a point)

and then conduct all possible operations and combinations using only first five, de facto doing what Euclid's Elements tried to show, and construct this perfect and applicable to our world geometry but don't use Euclid-Devac's 6th Axiom it does not mean that we have shown existence of God. It simply means that for our reasoning it is a redundant one, without saying actually anything either pro or against God's existence.

One of the examples of it is the Occams Razor.

Michał Heller (a Catholic priest, philosopher and a doctor of cosmology no less!) wrote quite a few books criticising such attempts and iirc his book Creative Tension perfectly shows his annoyance at presenting such proofs at one point. Seriously, if you will get an opportunity, give one of his books a try. Even if not for any other reason than to see a priest de facto bitch-slapping majority of low-effort attempts that come from religious people, similar to one I have presented here.

Logical fallacies for presenting arguments or refuting counterarguments

Maybe a lot of you didn't experience (or managed to forgot it, despite experience being an infuriating one. Although how you have managed that is beyond me) people dismissing your arguments with a counter like "What can you know, you are just a kid!" or "View like yours could result only from inexperience, so allow me to…". You see this? It was a logical fallacy that instead of refuting topic in question proceeded to attack the opponent but not his/her words. It's dirty, and you should be ashamed for deploying it in place of counterargument. "Because I'm older, therefore I'm right" is not an answer, it's a display of poor reasoning capabilities and personal culture.

Now to go down from my high horse for this one, however believe me that as someone who was always at least 2 years younger than his peers this made me detest quite a few 'older and therefore wiser' people with whom I had a displeasure to talk with.

Here are some most common examples (and here is a full list)

- Mentioned "attack the opponent not his words"

- Calling something an 'absurd' without explaining why

- Beginning from a thing we want to prove and use it to prove its correctness (also known as "circular reasoning" and shown by GB_Cobber in the discussion that I have linked above)