Is there a rule of thumb in English for which of two equivalent contractions to use [see examples]?

no rule that I can remember. The contraction at the front of the sentence tempts me to guess that you’re an engineer or technical writer. Funny, but I don’t think it would occur to me to state “I’ve not” instead of “I haven’t”. Are you fluent in other languages?

Unless you’re writing a paper (MLA master race), I wouldn’t worry.

Just going by what feels right, I usually tend to keep the first word of the sentence contraction-free.

“I’ve not…” is a more common expression in U.K. English than American English (where “I haven’t” is more common). As for the rest, there’s no reason that one would be a better choice than the other. Choose whichever sounds best to your ear.

Not fluent in anything else (I took some French in high school but I’m not great at it) and yes, a technical student. I’m not worried about this, just curious. :)

Equal weight grammatically to both. I agree with @dappled_leaves, as I usually do, about the UK usage of “I’ve not.”

There’s no rule of thumb grammatically because they’re all the same sentences, just contracted differently. But pragmatically, it can make a difference, depending on the way it’s spoken, the context, the relationship between the speaker and hearer.

In politeness theory, “It’s not allowed” is definitely more of a threat to negative face, which is when either the speaker or hearer has to submit to the will of the other. On the other hand “It isn’t allowed” is a lot more of a threat to positive face, which is when the speaker’s public reputation and well-being risks being damaged.

What really makes it possible to categorize it like this without the spoken context is that “isn’t” hides the “not” from the sentence – which is a word with a strong connotation in politeness theory.

@noservice So interesting, thank you.

@noservice Interesting “politeness theory” take!
@Mariah Before I read the post by @noservice I was going to say the preference would probably be regional. But the politeness theory angle gives a whole new outlook on contractions and the word “not”.

“It’s not allowed.” and “I’ve not gone there,” are more correct.
{Although I would prefer the 2nd sentence to read “I’ve not been there,”}

Both examples when written in full are exactly the same so why not write them that way.