How common is the usage of 'smth' in American English as an abbreviation of 'something'?
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Neither smth nor sth is a standard abbreviation in American English.
The first looks like Smith, which there’s surely no reason to abbreviate. Smithying, perhaps. The second looks like south, the direction. It also reminds me of my friend Seth, or perhaps in the right context the Sith from Star Wars.
So while I’m sure it means something, I can’t quite say what, and it’s a burden, even an imposition, on your readers to make them try to do so. I strongly advise against using sth to mean anything at all. Just type the thing out that you actually mean.
Nthr smth nr sth z a stndrd abrvshn n Amrcn Nglsh.
Th 1st lks lk Smth, whch thr’s shrly no rsn 2 abrvt. Smthyng, prhps. Th 2nd lks lk sth, th drcshn. It also rmndz me v my frnd Sth, o prhps n th rt cntxt th Sth frm Str Wrz.
So whl I’m shr it mnz smthng, I cn’t qt sy wht, & itz a brdn, evn an unknd impsshn, on yr rdrz 2 mk thm try 2 do so. I strngly advs agnst usng sth 2 mn anythng @ al. Jst typ th thng owt tht U actly mn.
In Taiwan, to teach verbs, smth and smb are used to shorten the explanation or definition. It's quite common here.
Smth is not used to my knowledge as a standard abbreviation, but I have seen sb for ‘somebody’ and sth for ‘something’ in idiom dictionaries such as the Oxford Idioms Dictionary and McGraw-Hill's Dictionary of American Idioms.