When did modern sheet music get invented/adopted by musicians?

The answer to this will depend on what you mean by "modern sheet music."

If you are talking about the development of music notation itself (let's say in a form that can capture a Mozart piano sonata), the answer is going to be "gradually over a span of almost a thousand years, beginning with the development of neumatic notation (basically showing the contour of a single line) around the 9th century and concluding [somewhat arbitrarily] in the mid-17th century with things like fractional time signatures (ie, 6/8) and 'advanced' accidentals like the double sharp." Throughout that time span, the history is obviously very rich and extremely messy, so it's hard to know what more to say about it here unless you want to ask something more specific (such as" how did we develop the capability to notate specific pitches?" or something). But if you are really looking for a comprehensive account of the whole history of notation, the best place to go would be the "Notation" article in the New Grove Dictionary of Music and Musicians, which is the standard academic Encyclopedia for the field. Another source worth your time would be Apel's The Notation of Polyphonic Music: 900-1600, despite its age.

If by "modern sheet music" you mean professionally printed and commercially available scores, as opposed to hand-written manuscripts, then the answer is a bit more specific. Obviously, Johannes Gutenberg's printing press in the middle of the 15th century provides the necessary technology to make printing happen in the first place. Musically, the most important early name is Petrucci, an Italian publisher who published a TON of music starting around 1500, mostly vocal repertoires in the form of "partbooks" (ie, one book per part, not complete scores showing all the parts together). I'm not quite as clear on exactly who the intended audience of these publications were. But with the exception of things like keyboard or lute music, most working musicians continued to play from hand-written scores until at least the early 19th century, and possibly later (I'm mostly aware of what 18th century musicians are doing, someone with more knowledge of other eras will have to say more on this point).

Are either of those things the kind of thing you are looking for? If not, I can try to answer more if you clarify what you mean a bit.