By definition, a logical fallacy is a form of reasoning that is both commonplace and inherently illogical.
Slippery slope
Slippery slope arguments are not inherently illogical. There's nothing inherently illogical about saying that one thing will lead to another. True, many slippery slope arguments are overly hasty or simply irrational, but that doesn't make "X will lead to Y" inherently illogical. By definition, a logical fallacy is inherently illogical. Therefore, "slippery slope" is not a logical fallacy. Particular slippery slope arguments may be flawed, but the correct response is not to shout "slippery slope fallacy!" but, rather, to explain why X won't necessarily lead to Y in those particular cases.
No true Scotsman fallacy
Appeal to authority
During my time as a philosophy instructor, I've seen this "fallacy" defined every which way: as thinking that X is true just because an authority said X; as citing an authority to support a position when the authority isn't a relevant expert; as relying on authorities rather than argumentation to back up a position; etc.
If we define the fallacy in the first way (as thinking that X is true just because an authority said X), then it is indeed a fallacy, but not one I'm too concerned about. Frankly, I wish most people would cite more authorities to back up their positions, as long as those authorities are relevant authorities. It's the citation of irrelevant authorities that I'm more worried about.
As for relying on authorities rather than one's own argumentation, whether that's objectionable depends to a large degree on the context. It isn't inherently illogical. Therefore, it is not a logical fallacy.
Naturalistic fallacy
The philosopher G.E. Moore
coined the term "naturalistic fallacy." He defined it as belief that just because everything with property X is good, property X must be the same thing as goodness. For example, suppose that Cary is right and an action is good insofar as it promotes universal happiness. In that case, someone would be committing the naturalistic fallacy if they concluded that the property of promoting happiness
is the same thing as the property of goodness.
What Moore described really is a fallacy, as far as I can tell. But nowadays people define "naturalistic fallacy" differently: as the belief that "natural" things tend to be better or healthier or whatever than "artificial" things.
Sorry, but that simply isn't a fallacy. There is nothing inherently illogical about the belief that natural things tend to be better. That belief may be false, just as string theory may be false. But if it is false, it is empirically false—false based on the evidence. Pure logic has nothing to say about whether natural sweeteners like stevia tend to be healthier than artificial ones like aspartame.
Appeal to emotion
This one really irritates me.
Properly defined, this is indeed a fallacy. That is, it's a fallacy to think that X is true just because X is emotionally pleasing. But hardly anyone who throws this term around uses it in that way.
I remember observing a critical thinking class in which the professor used the following as an example of the "appeal to emotion" fallacy: "Yes, I know that Ted is more qualified for the position, but Bob has been with the company for a long time and has always wanted that position. He's sacrificed so much for this company, and this is probably his last chance to get this promotion. Let's give Bob a chance." This may be a bad business decision, but there's nothing illogical about it. The speaker is perfectly clear about why they think Bob should get the job: they feel sorry for Bob and want to make him happy. If making Bob happy is the speaker's overriding priority, the conclusion "Let's give the position to Bob" is entirely logical. Calling it illogical because you have different priorities (such as maximizing profits) is like calling someone illogical for having a different favorite color. Giving Bob the job because you feel sorry for him may be unwise or even unethical, but it is not illogical.
Moving the goalposts
"Moving the goalposts" means repeatedly changing the standard of evidence that you demand. For example, suppose you refuse to believe that Michael Jackson had inappropriate interactions with kids and someone provides evidence that he did. You might demand a higher standard of evidence. If the other person manages to meet that higher standard and you then raise the standard even higher, your interlocutor might accuse you of "moving the goalposts." Defined this way, moving the goalposts is obnoxious and uncooperative but not a logical fallacy.
Let's first get something out of the way: there is nothing illogical about demanding a higher standard of evidence for claims that you dislike than for claims that you like. A demand is just that: a demand. It is not a statement. It does not have a truth-value. It does not stand in any logical or illogical relationships to other utterances. Heck, if you wanted to, you could demand lots of evidence for a claim that you already like and agree with, just for the hell of it. That isn't being "illogical." It's just being annoying. Logic enters the picture only if you actually believe that a statement is false just because you dislike the statement.
Now, if demanding a higher standard of evidence for claims that you
dislike isn't illogical, then neither is repeatedly demanding a higher
standard of evidence. Moreover, your interlocutor has no way of knowing what you actually believe unless they ask. For all they know, maybe you're just being a troll. Or maybe you demanded a relatively low standard of evidence because you thought their position was so absurd that they couldn't meet even that low standard, and now that they have somehow met the standard, you're raising the bar because you honestly believe that further evidence is necessary. If your interlocutors don't like it, they can give you the finger and walk away. No one's stopping them.
--Joseph