Is it Islamically preferable to fast the entire months of Sha‘ban and Muharram every year, or to fast most of them to achieve the utmost benefit, while breaking the fast on some days?

It is permissible to fast during Sha'ban, or most of it. The determination of the small portion that should be left unfasted is subject to custom. Therefore, it is not obligatory to fast the entirety of Sha'ban every year. It is preferable to conform to the Sunnah by refraining from fasting for a small number of its days. As for fasting the entire month of Muharram, there is no harm in it, and some of the Salaf did fast it completely. Nevertheless, it was not the guidance of the Prophet (peace be upon him) to complete any month other than Ramadan. Thus, it is preferable not to complete it, as indicated by the hadiths of Aisha and Ibn Abbas. Accordingly, fasting the entire month of Muharram is not a Sunnah.