Use rarer when you're comparing how uncommon two things are, and it sounds natural if the base adjective is short: This bird is rarer here than it used to be.
When a reader would write it
You're comparing two things and one is less common, so the comparative fits: Errors like that are rarer in the newer software. Use it when rare itself is the quality you're ranking, not rarity in general.
You're describing something over time, comparing then and now: Handwritten ledgers are rarer in offices than they once were. The word carries your comparison without needing more, and the two time points make the contrast clear.
You're drafting a formal review and want a slightly stronger feel than less common: Personal letters are rarer these days than emails. The shorter comparative keeps the sentence tight in measured, written prose.
You're not actually comparing anything, so hold off. This stamp is rare stands on its own; rarer only belongs once a second thing, time, or place enters the sentence and gives the comparison something to land on.
One qualifier that changes the choice
Stacking another intensifier is the situation worth checking first. A far rarer mistake these days is fine, and published writing also shows far more rare in similar spots, so treat both as acceptable. Pick whichever rhythm fits the rest of your sentence.
You're pairing two adjectives in one phrase. Scarcer and rarer keeps the comparison tight, and writers do use that shape. Some published sentences prefer the base form after adverbs like increasingly, so increasingly rare is also a natural reading. Both patterns appear in real usage.
Rarer is a real, usable word, not a slip. It's the standard short comparative of rare, and it shows up in published writing, as in Other elements are a little rarer, but their rarity doesn't always pay off. Trust your ear for register when choosing between it and more rare.