It is permissible to put the Holy Qur’aan on mobile phone, but it should be in the ‘Uthmaani script, unless that is not possible, in which case it is permissible to use a different one.
There is nothing wrong with carrying the Mushaf and putting it in one's pocket, in pants or other clothes, so long as it is protected against tearing or mishandling.
It says in Fataawa al-Lajnah al-Daa’imah (4/60): One of us carries the Mushaf in his pocket, and he may enter the washroom carrying it. What is the ruling on that, please advise us?
Answer: carrying the Mushaf in one's pocket is permissible, but it is not permissible for a person to enter the washroom carrying a Mushaf; rather he should put the Mushaf in a suitable place, out of respect and veneration for the Book of Allaah. But if he has no choice but to take it in with him, for fear that it may be stolen if he leaves it outside, then it is permissible for him to take it in with him, out of necessity. End quote.
But if the Mushaf is carried in the back pocket of one's pants, and that will lead to sitting on the Mushaf when a person wants to sit down, then it is not permissible -- in that case -- to put it in the back pocket. The least that can be said concerning that is that it is makrooh. In fact more than one of the scholars stated clearly that something less serious than that is haraam, namely putting the Mushaf beneath one's head like a pillow.
Al-Nawawi (may Allaah have mercy on him) said:
The scholars are unanimously agreed that it is obligatory to protect and respect the Mushaf.
Our companions and others said: ... it is haraam to use it as a pillow; in fact using any book of knowledge as a pillow is haraam. End quote.
Al-Tabyeen fi Adaab Hamalat al-Qur’aan (128); see also al-Burhaan fi ‘Uloom al-Qur’aan by al-Zarkashi (1/478)
Ibn Muflih (May Allaah have mercy on him) said:
It is makrooh to use the Mushaf as a pillow. This was stated by Ibn Tameem, who mentioned it in al-Ri’aayah. Bakr ibn Muhammad said: Abu ‘Abd-Allaah said it is makrooh to place the Mushaf beneath one's head and sleep on it. Al-Qaadi said: Rather that is makrooh because it is humiliating it and failing to respect it, as he is treating it like any other object.
Ibn Hamdaan favoured the view that it is haraam and stated that definitively in al-Mughni and al-Sharh. The same applies to all books of knowledge, if they contain Qur’aan, otherwise it is only makrooh.
Ibn ‘Abd al-Qawiy says in his book Majma’ al-Bahrayn: It is haraam to recline on the Mushaf and on the books of hadeeth and on anything in which there is any Qur’aan, according to consensus. End quote
Al-Adaab al-Shar’iyyah (2/393)
And Allaah knows best.