Hello EU
Here are some things that occur to me:
1. Discrepancies of register. A non-native speaker's English may be impeccable, for the most part; but slight failures of tone or register are most noticeable, in impeccable English. For instance, the non-native may use a word which a native speaker of that kind of English would never use in that context. Or a word may be used in a grammatical but unusual way. The commonest words are the most treacherous, in this connection: "quite", "nice", etc.
2. Absence of context. It's very difficult to provide sample sentences or chat for any length of time without revealing something about your background. When non-native speakers are writing naturally, they reveal something of their native background. When they are writing carefully, however, and perhaps do not wish to be taken for a native speaker, they reveal nothing. There may be literary references, for instance; but the little everyday details are missing.
3. Literary echoes. Sometimes non-native speakers use phrases they have come across in Shakespeare, Dickens, etc., or unwittingly catch the rhythm of well known writers.
4. Overly pure or consistent diction. By which I mean the kind of vocabulary we find in e.g. Jowett's translation of Plato. Non-native speakers who have reached a certain proficiency often dislike recent additions to everyday vocabulary, for instance.
5. Rhythm. Non-native speakers often bring a little of their native rhythm into English. The clauses may be not quite the right length; the pauses may sound mannered.
6. Grammatical regularity combined with inappropriate idioms. Grammar is the weakness of native speakers; idiom is the weakness of non-native speakers.
Maybe other members will have other ideas.
MrP