In a sheet of music there are several creative actions involved, and therefore several authors with copyright claims, and any of them can be in the public domain or not according to their date of death and (specially in the US) date of publication.
- The composer. Beware that the work of the composer is just the score he wrote.
- The arrangements. Beware that most classical pieces have been adapted and are even usually played with musical instruments that hadn't been invented by the time the music was composed. Arrangement creates separate copyright.
- The layout of the sheet. Even if the notes are just the same the composer wrote, usually changing it's graphic form creates a new copyright on the layout.
- Scanning. Scanning a public domain sheet might create new copyright rights for the person who did the scan in some countries (notably in the UK but not in the US).
In case of performance or recordings of a music piece, there are even more authors with copyright claims (the musicians, the conductor, the people who did the recording...). If you transcribe the recorded music, at least you will risk to infringe the copyright of the arrangement - and probably that of interpreters, too.
If you want to reuse a public domain composition the key point is that you need to identify what is in the public domain and then find a source for it, just to make sure that you are not mixing it with more recent (and copyrighted) material. A safe way would be to go for music scores old enough to be in the public domain - in the US I would go for music scores published before 1923 and in other countries I would search for sheets with editors dead more than 70 years ago (or 80, depending on the country).
If you take a recent sheet, it could be very difficult to tell apart what comes from the composer (and it's in public domain) from what has been added recently, but if you can, you can copy the public domain parts.
Anyway, I'd recommend trying to find scans of XIX century and early XX century sheets on the Internet.