No natural language in the world is completely and totally regular. We think of English as an extreme of irregularity, and it really is, but all languages have at least some part of their grammar where things don’t always go as planned. And there’s nothing wrong with that. That’s a natural part of a language’s evolution.
Conlangs, on the other hand, are often far too regular. For an auxlang, intended for clear communication, that’s actually a good thing. There, you want regularity, predictability. You want the “clockwork morphology” of Esperanto or Lojban. The problem comes with the artistic conlangs. These, especially those made by novices, can be too predictable. It’s not exactly a big deal—every plural ending in -i isn’t going to break the immersion of a story for the vast majority of people—but it’s a little wart that you might want to do away with.
Count the ways
Irregularity comes in a few different varieties. Mostly, though, they’re all the same: a place where the normal rules of grammar don’t quite work. English is full of these, as everyone knows. Plurals are marked by -s, except when they’re not: geese, oxen, deer, people. Past tense is -ed, except that it sometimes isn’t: go and went. (“Strong” verbs like “get” that change vowels don’t really count, because they are regular, but in their own way.) And let’s not even get started on English orthography.
Some other languages aren’t much better. French has a spelling system that matches its pronunciation in theory only, and Irish looks like a keyboard malfunction. Inflectional grammars are full of oddities, ask any Latin student. Arabic’s broken plurals are just that: broken. Chinese tone patterns change in complex and unpredictable ways, despite tone supposedly being an integral part of a morpheme.
On the other hand, there are a few languages out there that seem to strive for regularity. Turkish is always cited as an example here, the joke being that there’s one irregular verb, and it’s only there so that students will know what to expect when they study other languages.
Conlangs are a sharp contrast. Esperanto’s plurals are always -j. There’s no small class of words marked by -m or anything like that. Again, for the purposes of clarity, that’s a good thing. But it’s not natural.
Phonological irregularity
Irregularity in a language’s phonology happens for a few different reasons. However, because phonology is so central to the character of a language, it can be hard to spot. Here are a few places where it can show up:
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Borrowing: Especially as English (American English in particular) suffuses every corner of the planet, languages can pick up new words and bring new sounds with them. This did happen in English’s history, as it brought the /ʒ/ sound (“pleasure”, etc.) from French, but a more extreme example is the number of Bantu languages that borrowed click sounds from their Khoisan neighbors.
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Onomatopoeia: The sounds of nature can be emulated by speech, but there’s not always a perfect correspondence between the two. The “meow” of a cat, for instance, contains a sequence of sounds rare in the rest of English.
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Register: Slang and colloquialism can create phonological irregularities, although this isn’t all that common. English has “yeah” and “nah”, both with a final /æ/, which appears in no other word.
Grammatical irregularity
This is what most people think of when they consider irregularity in a language. Examples include:
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Irregular marking: We’ve already seen examples of English plurals and past tense. Pretty much every other natural language has something else to throw in here.
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Gender differences: I’m not just talking about the weirdness of having the word for “girl” in the neuter gender. The Romance languages also have a curious oddity where some masculine-looking words take a feminine article, as in Spanish la mano.
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Number differences: This includes all those English words where the plural is the same as the singular, like deer and fish, as well as plural-only nouns like scissors.
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Borrowing: Loanwords can bring their own grammar with them. What’s the plural of manga or even rendezvous?
Lexical irregularity
Sometimes words just don’t fit. Look at the English verb to be. Present, it’s is or are, past is was or were, and so on. Totally unpredictable. This can happen in any language, and one way is a drift in a word’s meaning.
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Substitution: One word form can be swapped out for another. This is the case with to be and its varied forms.
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Meaning changes: Most common in slang, like using “bad” to mean “good”.
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Useless affixes: “Inflammable means flammable?” The same thing is presently ongoing as “irregardless” becomes more widespread.
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Archaisms: Old forms can be kept around in fixed phrases. In English, this is most commonly the case with the Bible and Shakespeare, but “to and fro” is still around, too.
Orthographic irregularity
There are spelling bees for English. How many other languages can say that? How many would want to? As a language evolves, its orthography doesn’t necessarily follow, especially in languages where the standard spelling was fixed long ago. Here are a few ways that spelling can drift from pronunciation:
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Silent letters: English is full of these, French more so. And then there are all those extra silent letters added to make words look more like Latin. Case in point, debt didn’t always have the b; it was added to remind people of debitus. (Silent letters can even be dialectal in nature. I pronounce wh and w differently, but few other Americans do.)
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Missing letters: Nowhere in English can you have dg followed by a consonant except in the American spelling of words like judgment, where the e that would soften the g is implied. (I lost a spelling bee on this very word, in fact, but that was a long time ago.)
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Sound changes: These can come from evolution or what seems like sheer perversity. (English gh is a case of the latter, I think.)
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Borrowing: As phonological understanding has grown, we’ve adopted a kind of “standard” orthography for loanwords, roughly equivalent to Latin, Spanish, or Italian. Problem is, this is nothing at all like the standard orthography already present in English. And don’t even get me started on the attempts at rendering Arabic words into English letters.
In closing
All this is not to say that you should run off and add hundreds of irregular forms to your conlang. Again, if it’s an auxlang, you don’t want that. Even conlangs made for a story should use irregular words only sparingly. But artistic conlangs can gain a lot of flavor and “realism” from having a weird word here and there. It makes things harder to learn, obviously, but it’s the natural thing to do.