According to an annotated version of the FIDE rules (FIDE Laws of Chess 2017, by the Chess Arbiters' Association), it's banned in some tournaments, which presumably means it's not banned in all tournaments. The relevant annotation (the bottom of p.23) says:
Article 11.3.1 has resulted in some tournaments banning the use of scorebooks. Where these are allowed arbiters should ensure that players do not refer to earlier games.
I've never played in a FIDE tournament, but I've played in USCF tournaments. The relevant USCF rules are similar to the FIDE rules. A tournament can prohibit using scorebooks, but in my (amateur) experience they don't; probably many tournament directors are happy to not have to provide as many scoresheets.
USCF rule 15A (similar to FIDE rule 8.1.1) states that you are to record your moves "on the scoresheet prescribed for the competition". This is clear, but it's also only relevant if the competition prescribes a scoresheet.
USCF rule 20B (simiar to FIDE rule 11.3.1) states you are forbidden to "make use of" notes or recorded matter - it doesn't prohibit having them in a place where you can't see them. Generally a notation booklet will be open in such a way that you only have the current game visible, and not any previous games. Browsing notes on previous pages would of course be cheating, but it would also be fairly obvious to your opponent and any bystanders. The only legitimate reason to flip a page after the game has started is if you get to move 50 or so, and by that time any opening notes would be rather useless.
While USCF rule 15G (similar to FIDE rule 8.3) does state that the scoresheets are property of the sponsoring organization, I've never had a tournament ask me to rip a page out. I get the feeling that this rule is mostly there so a player can't refuse to show it during a dispute, and it allows the arbiter/TD to insist that the opponent be able to use it to fill in his own sheet.
And finally, the USCF sells these score books on their website, calling them "A must for the 'organized' tournament player!". Presumably, they would not do so if it was ordinarily illegal to actually use them for their intended purpose.