Monserrat is a variant spelling of the Catalan name Montserrat. It is likely the better spelling in English-speaking countries given the correct pronunciation of the name. After appearing on the US Top 1000 popularity list each year from 2000 to 2009, it reentered the list in 2014 at Number 592 and now ranks Number 766.
Monserrat/Montserrat is a Catalan name meaning "serrated mountain." That may not be the most appealing meaning, but the exceptional beauty of the Montserrat mountain range near Barcelona, Spain might change your mind. Monserrat is, unsurprisingly, most common among the Spanish community. Montserrat is also one of the claimed homes of the Holy Grail according to Arthurian legends.