Substantial obscurities about American criminals (9)

'criminals' is the definition.
The answer and definition can be both people as well as being plural nouns.
Perhaps there's an association between them I don't understand?

'substantial obscurities about american' is the wordplay.
'substantial' becomes 'big' (I've seen this in another clue).
'obscurities' becomes 'mists' (I can't justify this - if you can you should give a lot more credence to this answer).
'about' means one lot of letters go next to another.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'mists' put after 'a' is 'amists'.
'big'+'amists'='BIGAMISTS'

Can you help me to learn more?