1. What does it mean?
모솔 is short for 모태 솔로, an exaggerated expression for having been single since birth. It usually refers to having no past dating relationship rather than simply being single right now.
Also written or said as 모태 솔로, 모태솔로, motae solo
2. The nuance
The phrase may be a joking self-description, but applying it to another person can embarrass or belittle them. Someone's relationship history is private, and being single is not a defect. The label also does not tell you whether a person wants to date.
3. Examples
“나는 모솔이라 소개팅이 어떻게 진행되는지 궁금해.”
I've never dated, so I'm curious about what a blind date is like.
“친구의 연애 경험을 묻고 모솔이라고 놀리는 건 싫어.”
I don't like prying into a friend's dating history and teasing them for never having dated.
Sources
KoreanClass101: Korean language lessons
References support the meaning and usage. Example sentences are written for NoCapWiki.