Which is more commonly used to say twenty-two?
veinte y dos
I know for numbers above 30 that the first example is always the case. I am also wondering about 16-19.
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The correct form is veinti-. The Real Academia de la Lengua Española (Royal Achademy of Spanish Language), RAE, publishes this and more things on its web page.
A list of numbers, where your answer is, is in:
(numerales cardinales) http://lema.rae.es/dpd/srv/search?id=rqV8h362gD62vc21qB
Which comes from:
All about numbers in spanish: http://lema.rae.es/dpd/srv/search?id=QHaq7I8KrD6FQAyXTS
All this information is in the DPD -Diccionario panhispánico de dudas (Panhispanic dictionary of questions). Try to check it whenever you have a question, it's very useful. Also the DRAE dictionary will help you very much.
Veintidós is by far the most common as that matches the pronunciation of (as far as I know) every single Spanish-speaker. It used to be acceptable to say veinte y dos and you'll probably see it still once in a while, but as of the 2002 DPD, they are considered antiquated and recommended to be avoided though not strictly incorrect.1 It's curious to note that the 22nd DRAE defines veintidós as veinte y dos, but in accordance with the DPD's recommendations, the 23rd DRAE changes that to veinte más dos.
Writing diez y seis in separate words is even more outdated because the diez component is at this point completely destressed.
Also, numbers above 30 may be written as a single word, just make sure you don't omit the -a: treintaicinco, sesentaisiete, noventaiún/uno/una.
1. I would only recommend it if you're a native and actually pronounce the e in veinte and/or realize two separate fully stressed syllables as in véinte y siéte. These days, most everyone destresses the first syllable (veinte y siéte) and raises the e (veinti ̮y siete), both of which are cause for merging into a single word. If you do either, or if you are a non native, you should only ever consider using the single word.
veintidós (22) is correct, veinte y dos is not. It is written in a single word (always) from dieciséis to veintinueve.