What are the popular hadiths that are not authentic, and the lesser-known hadiths that mention great virtues and rewards?

Unsound hadiths and sound hadiths unknown to many are widespread. Among the unsound hadiths are: "Whoever's prayer does not restrain him from indecency and evil, he will not increase in relation to Allah except in remoteness" (false), and "The good deeds of the righteous are the evil deeds of those brought near" (false, has no basis), and "The month of Ramadan is suspended between the heavens and the earth and is not raised to Allah except with Zakat al-Fitr" (weak), and "Paradise is beneath the feet of mothers; whoever they wish, they admit, and whoever they wish, they turn away" (munkar).

Among the sound hadiths are: - Reciting "Say: He is Allah, the One" is equivalent to one-third of the Quran. - Great virtue for one who washes thoroughly and performs ghusl, goes early to the mosque on Friday, listens, and does not engage in idle talk. - Saying "Glory be to Allah, the Most Great, and praise be to Him" plants a palm tree in Paradise. - A dirham given as sadaqa by a poor person is better than one hundred thousand dirhams given by a rich person.

One can refer to Al-Albani's books, such as Silsilat al-Ahadith as-Sahiha and ad-Da'ifa, and the Dorar website for more information.