Is swearing "by the Prophet" considered a vain oath for which there is no expiation, given its common usage, or is it impermissible to say it at all because it is polytheism, even if the tongue has become accustomed to it?

Swearing by other than Allah, including swearing by the Prophet or the Kaaba, is not permissible according to the preponderant opinion; in fact, it has been said to be polytheism (shirk). It is not considered a binding oath that requires expiation (kaffara) when violated, even though some scholars differed and held that expiation is obligatory. This type of oath is unrelated to the false oath (yamin al-ghamūs) or the idle oath (yamin al-laghw). Whoever is afflicted with it must abandon it and guard their tongue. Jurists disliked or prohibited swearing by other than Allah. If one swears by other than Him, their oath is not binding. While there is a difference of opinion regarding the ruling on swearing by other than Him if one is truthful, if one is lying, it is definitively forbidden, in order to preserve the sanctity of revered religious symbols.