I know about the "standard" special katakana, like ファ fa, ディ di, ヴ vu, etc. However, I keep running into others I can't find on any list, including the ones in What special kana are used to write foreign words?
For example, on this Wikipedia page, one character has the name:
Quark (クォーク Kwōku)
I've never seen クォ kwo in any katakana list.
Another example is フュ fyu, as in the new movie X-MEN: フューチャー&パスト.
So, what are all the "non-standard" katakana in regular use?
Here is a list of extended katakana color coded based on usage. In my personal experience, I have seen the orange ones and most of the blue ones in actual use multiple times. The beige and purple ones I have never seen used before (except maybe スィ), so don't bother learning them.
Here's a list with example words I made based on the link, roughly ordered by commonality based on my experience and the amount of noteworthy dictionary hits:
Common extended katakana:
Uncommon extended katakana:
Rare extended katakana:
フュ (fyu) ― フューチャー (future)
ヴァ (va) ― ヴァイオリン (violin), ヴァンパイア (vampire) (more commonly バイオリン and バンパイア)
ヴォ (vo) ― ヴォーカリスト (vocalist) (more commonly spelled ボーカリスト)
ドゥ (du) ― シルク・ドゥ・ソレイユ (Cirque du Soleil), ドゥーイットユアセルフ (do-it-yourself)
(even when spelled with ヴァ, ヴィ, ヴ, ヴェ or ヴォ, the words are often still pronounced with a B sound rather than a V. They just look more fancy or foreign or whatever.)
Almost exclusively in foreign names:
Other combinations such as スィ (si) and ホゥ (hu) also exist, but unlike the ones listed above they are often substituted for other katakana. For example, simple is シンプル and hood is フード. But they can at times be used for portraying foreign sounds more accurately. Most often in names of people and places, or transcriptions of foreign words that are not used as loan words in a Japanese sentence. (Very similar to the ヴァ row, except those are much more commonly used for alternate spellings of loan words)
Lastly, as mentioned by OP in the comments, it's worth mentioning small vowel kana (ぁ, ぃ, ぅ, ぇ, ぉ) can be used for long vowels, a practice that I believe is actually more common in hiragana. It's almost a different matter entirely, but for completion's sake I'll explain it anyway.
If a small vowel kana is used after a kana with the same vowel (e.g. きぃ, ねぇ, ハァ, ルゥ) it means it's a long vowel (similar to きい, ねえ, ハー, ルー). I have a hard time finding a source that explains the nuances implied by use of a small vowel kana, but from personal experience I know four uses:
I think that there may not be any answer to this question - at least not a good one that includes objectively defining "regularity" without any context. Surely, some words including some combination(s) are especially common, looking at Japanese as a whole, and I'd hazard (from a subjective standpoint) that combinations with "ウ + small version vowel" as well as ティ are among those, but beyond that? Well, according to Wikipedia:
Small versions of the five vowel kana are sometimes used to represent trailing off sounds (ハァ haa, ネェ nee), but in katakana they are more often used in yōon-like extended digraphs designed to represent phonemes not present in Japanese; examples include チェ (che) in チェンジ chenji ("change"), and ウィ (wi) and ディ (di) in ウィキペディア Wikipedia.
I believe that this phenomenon could easily (but roughly) be categorized into two usage cases. The first would then be loanwords, where パーティー is a good example.
Slightly off topic, I think it's worth mentioning that while English phonemes not present in Japanese can be rather easily identified, the "rules" for spelling in katakana may be harder to discern. Looking at how "party" becomes パーティー with katakana, an unknowing student may be tempted to write "ticket" as ティケット, but alas, it's チケット. Certainly, there are tendencies (is it a leading, trailing, middle, surrounding phonemes etc.), but there is also some amount of discretion involved. (Why is a word beginning with [ˈka] or [ˈkæ] that could arguably be spelled カ as in カメラ, suddenly キャ, as in キャンパス?)
The second usage case I was thinking about is proper names. Naturally, there are many "fixed" spellings of given names for people, countries and such, for example ウィリアム, or スウェーデン to name but a few. The desired pronunciation, and hence katakana spelling, in the case of people may however be a matter of individual preference.
And by the way, not all loanwords are English in origin, of course. Even some that are easily mistaken for "English" are not, like タバコ (Portuguese), ビール (Dutch) and アレルギー (German). Sounding rules of the source language may also affect how a word is transcribed.
For example, my own middle name is Knut. In my native language the "k" is "hard", and the "n" is pronounced separately, giving roughly [kn'ʉːt] (in contrast to words like "knight" in English). When having this name transcribed for my Japanese student's ID (many years ago), they went with the English sounding rules, in this case giving ナット, which to me was... Not perfect.
Anyway, if you were to construct a list of foreign words in Japanese that included "extended digraphs", then took it upon you to study their frequency, then you'd probably find that some are rather more common then others, but as I've been saying, that's not all to consider. E.g. in the context of mahjong, where most loanwords are Chinese, you'd find both high frequencies and a practical use for one set of "extended digraphs" different from what is common in another.
Absent a context, "anything goes". If your question would have been "which extended katakana are there", many dictionaries and such include lists. Googling, I found this one on-line, that contains I think all combinations that I've ever seen, and some that I haven't seen before:
Katakana (including List of extended katakana)
I hope this helps.
Edit: Note that the chart in the list linked above, and others linked in other answers, may change from time to time.
I'm a Japanese student (so, I am not good at speaking in English, sorry for my broken English)
Historical, people used to regard Katakana as Men's words. and Hiragana was seemed as Women's words.
above is a part of Old Japanese Constitution "大日本帝国憲法" announced in 19century
第１章 天皇
第１条大日本帝国ハ万世一系ノ天皇之ヲ統治ス
There isn't any Hiragana because there was thought that publish writings should not be use women's words.
NOWADAYS, there is no difference in gender between Hiragana and Katakana.
so, Is there difference between Hiragana and Katakana?
Answer is "Yes"
At present, Hiragana is used in Japanese ordinal words like "sashimi","Tsunami","Kawaii" (Japanese ordinal words are called "Wago"「和語」 in Japanese)
and, Katakana is used in expressing foreign language, expect for archaic Chinese words like Wikipedia(from English) as ウィキペディア, Chocolat(from French) as ショコラ in Japanese.
10/24 PM6:00(in Japan Time) added It is difficult for "non-standard katakana" to define. Nowadays, new non-standard katakana is created. But there is a rule.
Standard Katakana
(a consonant)+ya→ャ ex. キャラ(kyara)
(a consonant)+yu→ュ ex. キュート(kyuto)
(a consonant)+yo or ho→ョ ex.ショック(syokku/shokku)
Non Standard Katakana
(a consonant)+wa→ｧ ex.かぁちゃん(kwachan) (It means a mother. but informal expression)
(a consonant)+(weak)i→ィ ex.カービィ(ka-b (weak)i) ,フレンドリィ(furendor(weak)i) but ｨ sometimes replace - instead of i, except for a proper noun.
(a consonant)+(weak)fu→ｩ this is a sound of wind for Japanese ex.フゥーと(fuu-to)
(this ONOMATOPE is sound of blowing candles out)
But Words from foreign languages isn't used ｩ, generally use ｭ ex.フュージョン(fu-zyon/fyu-zyon)
(a consonant)+we or he→ｪ ex.クウェート(kwe-to), シェパード(shepa-do)
(a consonant)+wo→ｫ ex.クォーク(kwo-ku)
Thank you for reading
