Verbatim · Ayatollah A. Sistani
Yes, she can read the Quran except for the those verses which include obligatory Sajdah, and also she cannot touch the scripts of the Quran. She can listen to the Quran, including the verses of obligatory Sajdah, and if they listen to the verses of obligatory Sajdah, they must perform Sajdah
Ayatollah A. Sistani sistani.org Q&A #14835 other marjas may differ
A. Sistani’s office answers all three possible questions in one. Reading is permitted except the four verses of obligatory sajdah. Touching the script is forbidden. Listening is permitted, even to obligatory-sajdah verses, with the caveat that the woman must perform sajdah immediately upon hearing one.
The four obligatory-sajdah verses are Surat al-Sajdah (32:15), Surat Fussilat (41:37), Surat al-Najm (53:62), and Surat al-Alaq (96:19).
See the longer companion answer at quran-during-period-verses for the explicit list, and listen-sajdah-during-hayd for the listening-specific ruling.