For two well-known examples of this in English, it’s common to say/hear
“Me and my friend” or “Him and I” (both incorrect)
instead of
“My friend and I” and “He and I” (both correct).
Similarly, I'm wondering if there are any (and if so, what they would be) prime, well-known examples of non-grammatical French structures that, while "technically wrong,” are so commonly (mis)used by French people that they are considered by many, if not most, to be "basically correct"?
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Yes, there are, as in all languages, see for example:
Malgré que
Or pleonasms such as:
Au jour d’aujourd’hui
However, note that they are incorrect even if you may sometimes hear them. They should be seen as common mistakes.
More explanation about "Au jour d’aujourd’hui" (Thanks comments! source: academie-francaise)
Emplois fautifs
Cette formule jouant sur la redondance a une résonance familière et plaisante. Pour parler de la journée présente ou de celle que l’on évoque, on préfèrera dire Aujourd’hui ou Ce jour. On peut aussi recourir à la tournure ancienne Ce jour d’hui.
Many people use the subjonctive after "après que". We have to use the indicative. e.g.:
Wrong: Après que je sois allé chez Jules.
Correct: Après que je suis allé chez Jules.
