Sayyida Umm Salama (may Allah be pleased with her) said: I did not see the Messenger of Allah (peace and blessings be upon him) fasting an entire month except for Sha'ban. He used to join the fasts of Sha'ban with Ramadan so that it would be as if he fasted two consecutive months. He would fast during a month to the extent that we would say: Now he will not break his fast, and then he would break his fast to the extent that we would say: Now he will not fast.
In this hadith, there is mention of the Prophet Muhammad sallallahu alayhi wa sallam fasting the entire month of Sha'ban, but in Bukhari and Muslim, there is a hadith from Sayyiduna Abu Hurairah radi Allahu anhu prohibiting fasting the entire month of Sha'ban, which takes precedence over this hadith.
Because when there is a conflict between a statement and an action, the statement is given precedence. In one narration, it is stated that if anyone tells you that the Messenger of Allah sallallahu alayhi wa sallam fasted the entire month other than Ramadan, do not affirm it.
Similarly, in the hadith of Sayyidah Aisha radi Allahu anha (1514), it has already been mentioned that the Prophet sallallahu alayhi wa sallam did not fast the entire month of any month other than Ramadan.
Further ahead, in hadith number (1781), it is also narrated in the same way from Sayyiduna Ibn Abbas radi Allahu anhuma.
Therefore, one should not fast after the 15th of Sha'ban, and Allah knows best.
Caution lies in this: that after the 15th of Sha'ban, one should not fast until the beginning of Ramadan, as will be mentioned ahead.