Nisaa
The ruling depends on exactly what she is touching: the Arabic pages of a printed Mushaf, its cover, a translation without the Arabic text, or a phone screen. These are treated differently, and it helps to take them one at a time.
The majority of scholars, across all four madhabs, hold that a menstruating woman may not touch the Arabic text of a Mushaf directly. Their evidence is the ayah (interpretation of the meaning): "None shall touch it except the purified." [Al-Waqiah 56:79]. They also cite the letter the Prophet (Salla Allahu 'Alayhi Wa Sallam) sent to the people of Yemen through Amr ibn Hazm, which stated: "No one should touch the Qur'an except one who is pure." (Narrated by Malik in Al-Muwatta. The chain has a gap, but al-Shafi'i and later scholars such as Ibn Abd al-Barr accepted it as supported by other routes, and it is what the majority of scholars act on.)
A minority of early scholars, among them al-Hasan al-Basri, Muhammad ibn Sirin, and Ibn Hazm (the later Andalusian jurist, not the companion Amr ibn Hazm above), held that touching the Mushaf without purity is permitted. They read the ayah as describing the Preserved Tablet (al-Lawh al-Mahfuz), which only the angels touch, rather than as a ruling about the earthly copy in a woman's hands. The four madhabs did not adopt this view, but it is a real, named position, not a fringe one.
Touching the Mushaf through something separate from it, such as the bag it is carried in, a sleeve, or a glove, is agreed across the schools to be permitted. She is touching the barrier and not the Mushaf itself, so the restriction listed in what is not allowed during hayd does not reach that far.
The Maliki school adds a further concession: a woman who is teaching or learning Qur'an may touch and recite from the Mushaf during hayd or nifas out of necessity, since going without it for the length of her bleeding would disrupt her teaching or her studies. This is specific to teaching and learning, not a general Maliki permission to handle the Mushaf freely.
A Qur'an translation with no Arabic text is where the four madhabs genuinely part ways. The Hanafi school treats the Qur'an as its wording and its meaning together, so a translation still counts as Qur'an and the same restriction applies. The Maliki, Shafi'i, and Hanbali schools hold that the Qur'an's inimitability is specifically in its Arabic wording, so a translation on its own, without the Arabic, may be touched without purification.
Reciting from memory or listening to Qur'an involves no touching at all, so it falls under a different ruling. See can you recite Qur'an during your period for that separate difference of opinion, one of several acts of worship that stay open to her during hayd. The same distinction, touching versus reciting, holds for Janabah too; see can you recite Qur'an during Janabah for how that question is decided.
A phone, tablet, or Qur'an app is the version of this question most women are actually asking, and it is new enough that no classical text addresses it directly. Contemporary scholars have reasoned it from the same principle behind the barrier exception above: what the evidence restricts is touching a Mushaf, and a phone is not one, since its screen is glass and plastic showing changeable pixels, not the fixed Arabic text a Mushaf is defined by. Islamweb's fatwa council put it in exactly those terms (fatwa no. 465004, 17 Rabi al-Awwal 1444 / 12 October 2022): purification is not required to hold the device, open the app, or scroll through it. The fatwa still asks for basic respect, such as not reading from a screen that is visibly unclean. This is now the position of most contemporary scholars and fatwa councils across the schools, whether the state is hayd, nifas, or janabah.
The same barrier and app exceptions travel with her on Hajj or Umrah too; see what if your period starts during Hajj for what else changes, and what does not, during the pilgrimage.