According to a report from ʿĀʾishah (may Allah be pleased with her), the month in which the Messenger of Allah ﷺ fasted the most after Ramaḍān was Shaʿbān. In one narration—understood to have occurred in some years rather than consistently—it is reported that he fasted the entirety of Shaʿbān.
(al-Bukhārī, Ṣawm, 51; Muslim, Ṣiyām, 175–176)
Apart from this, there is no specific fast known as the “fast of the Three Sacred Months.” However, it is commendable for a believer to make good use of the blessed period known as the Three Sacred Months—namely, Rajab, Shaʿbān, and Ramaḍān—by increasing acts of worship that draw one closer to Allah, including frequent fasting. This is the proper way to benefit fully from this season of spiritual opportunity.
If a person has no valid excuse that prevents fasting, they are already obligated to observe the entire fast of Ramaḍān. If they wish, they may also fast all of Rajab and Shaʿbān. However, extending this practice under the designation of a distinct “Three Sacred Months fast,” generalizing it to all people, or treating it as a specific act of worship akin to the Ramaḍān fast would require explicit evidence from the Qurʾān and the Sunnah, and no such evidence exists.