How irregular!
I often say that the problem most people have with grammar isn't the grammar itself, but how it's described (I even wrote a post about this a couple of months ago).  The Romans came up with quite a sophisticated way to describe grammar, and it was so successful that we still use it to this day.  However, what a lot of grammarians still haven't twigged is that what meant a lot to Romans means absolutely nothing to your average inhabitant of 21st century Earth.
One of the words that would be completely straightforward to a Roman is regular, and of course the converse irregular.  People familiar with grammatical terminology tend to think it should be easy for an English speaker too, because the root of the word is so common in English: reign, rule, regulations.  A "regular" form is one that follows the rules.  Simple.
Except it's not, because we don't use the word regular to mean a rule-follower in any other situation.  Outside of language circles, it means to do something with a predictable frequency or schedule.  So what - it's a different thing, so there's no need for confusion, right?  Wrong, and this is a subtlety that's easy to miss, even though it goes to the very heart of irregular forms.
The majority of words in any language are regular -- the vast majority follow the rules.  The irregular ones, the ones that break the rules, are in a minority.  Which words are irregular?  Well, it's always the common ones: to be, to have, to go; child/children etc.  This is uncontroversial - it's a well-known statistic.  That there are a few uncommon irregular forms (eg ox/oxen) doesn't break the rule, because these are forms that were common relatively recently (oxen were still in use 100 years ago, because not everyone could afford a new-fangled "tractor") and are being lost anyway (when did you last call talk about an "ox"?).
So here we have a rather nasty piece of cognitive dissonance - the forms that are most regular in terms of frequency of occurrence are the ones we call irregular, and the ones that are least regular by frequency of occurrence are called regular.
A word that is supposed to help us understand actually ends up confusing us further, and we're not even sure why we're confused.  Not good.
This problem isn't limited to English, though, as the equivalent word in the Romance languages (Italian, French, Spanish etc) tends to have a similar meaning to the English.
The failure to understand the concept of regular has profound consequences in the teaching of forms, particularly when it comes to verbs.
How irregular?

The first thing that people forget is that regularity is not a yes/no question.  While the majority of verbs are completely regular, some "irregular" verbs are only slightly irregular.

For example, the conditional and future simple of a Spanish verb are formed by adding a suffix to the infinitive.  There are no verbs in the language that are irregular in terms of the suffixes.  There are a handful of verbs that don't use the infinitive, instead forming a "future stem" by dropping the vowel from the infinitive ending.  But then, is this even an irregularity?  The process we're looking at here has a name -- syncope -- and it occurs in other languages.  You can argue, then, that these future forms aren't irregular, because they do indeed follow a rule.  After all, the 3 major verb groups in Spanish (-ar, -ir, -er) all follow different rules, yet we still refer to "regular" verbs in each "conjugation".  So if we make a category of "vowel-dropping verbs", suddenly we find that we've defined the Spanish future as having no irregular verbs whatsoever.  At the very least, I would argue that the verbs that undergo syncope are only slightly irregular.

But maybe it's unfair of me to talk about the future stem, because as a rule Spanish stems are far more stable than their counterpoints in Italian and French.  In all three languages, the conjugation of "to go" is built on three different Latin roots -- ire, andare, vadare -- making it the single most irregular verb in the language (whereas the most irregular verb in English is be/is/was).  But with other irregular verbs, Spanish picks a stem for a tense and runs with it.  So while the present tense of "to have" in French and Italian is bisyllabic in the 1st and 2nd plural forms but monosyllabic in the singular forms, both words in Spanish ("tener", lexical verb; "haber", auxiliary verb) stay consistent across all persons, even though these are irregular verbs.

But all in all, we can see that some verbs are more irregular than others.

Introducing...

But if there is a scale of irregularity, what is the extreme of this scale?  I'd like to introduce a new term to describe it.  I call it...

...Contraregular

There are words out there that completely ignore the rules (eg go -> went), but there are others that go completely against the rules.

For example, nouns ending in -o in Spanish and Italian are masculine, as a rule (rule -- Latin regula -- regular).  Yet "mano" (hand) is feminine.  It goes completely contrary to the rule, hence "contraregular".

Why is this important?

This goes back to the fundamental nature of irregular forms that I mentioned earlier: they only occur in frequent words or structures.  And in fact, the most common items are generally the most irregular.

And what is it that we tend to teach first?  That's right, the most frequent words and structures.  Hence the most irregular -- including the contraregular.

So in Spanish, you might learn the following within the first hour:
Good morning - Buenos días
Good afternoon/evening - Buenas tardes
Good night - Buenas noches

These are all contraregular -- -es normally marks masculine plural, but tarde and noche are feminine nouns, so the adjectives are marked in feminine plural. -as normally marks the feminine plural, but día is actually masculine, so the adjective is marked in masculine plural.

This isn't that big a deal if you just tell someone about it, but in many classes, you don't -- the student is expected to infer the grammar from examples.  If your students are first exposed to counter-examples, to exceptions, how can they generalise?  The irregular forms become a blockage, and the students are forced to learn each example as if there were no rules whatsoever.

But even if we do explain, should we be teaching this before we've covered the basics of regular adjectives?  I see no reason why we should.  There is no proven pedagogical advantage to being able to parrot a few fixed phrases before learning to use the grammar productively.  As a practical matter, such greetings may seem immediately useful, but there is no genuine value in being able to say "good morning" when you are otherwise incapable of saying anything in the language.  Besides, within a few hours in the classroom, you should be able to get your students to the point where they can construct such phrases themselves with a little bit of guidance.