In Summary 
It is unlawful (haraam) for someone not in the state of wudu to carry a Qur'an, even by a trap or in a box , or touch it, whether its writing, the spaces between its lines, its margins, binding, the carrying strap attached to it, or the bag or box it is in.  

The opinion expressed in Fiqh al-Sunnah that it is permissible to touch the Qur'an without ritual purity is a deviant (shadh) view contrary to all four schools of jurisprudence (fiqh) that is Hanafi, Maliki, Shafi'i and Hanbali and is impermissible to teach (except to teach that it is deviant).  

It is permissible to carry books of Sacred Law (Shari'ah), hadith, or Qur'anic tafsir, provided that most of their text is not Qur'an.  

One should not use saliva on the fingers to turn the pages of the Qur'an.  

When one fears that a Qur'an may burn, get soaked, that a non-Muslim may touch it, or that it may come into contact with some filth, then one must pick it up if there is no safe place for it, even if one is not in the state of wudu or is in need of the obligatory bath, though performing the dry ablution (tayummum) is wajib if possible.  

It is haraam to use the Qur'an or any book of Islamic knowledge as a pillow.



Imam Muhammad ibn Ahmad Qurtubi says in  al-Jami' li ahkam al-Qur'an [Taken from Reliance of the Traveller]

It is the inviolability of the Qur'an: 

1. not to touch the Qur'an except in the state of ritual purity in wudu, and to recite it when in a state of ritual purity;
2. to brush one's teeth with a toothstick (siwak), remove food particles from between the them, and to freshen one's mouth before reciting, since it is the way through which the Qur'an passes;
3. to sit up straight if not in prayer, and not lean back;
4. to dress for reciting as if intending to visit a prince, for the reciter is engaged in an intimate discourse;
5. to face the direction of prayer (qiblah) to recite;
6. to rinse the mouth out with water if one coughs up mucus or phlegm;
7. to stop reciting when one yawns, for when reciting , one is addressing one's Lord in intimate conversation, while yawning is from the Devil;
8. when begining to recite, to take refuge from in Allah from the accursed Devil and say the Basmala, whether one has begun at the first surah or some other part one has reached;
9. once one has begun, not to interrupt one's recital from moment to moment with human words, unless absolutely necessary;
10. to be alone when reciting it, so that no one interrupts one, forcing one to mix the words of the Qur'an with replying, for this nullifies the effectivness of having taken refuge in Allah from the Devil at the beginning;
11. to recite it leisurely and without haste, distinctly pronouncing each letter;
12. to use one's mind and understanding in order to comprehend what is being said to one;
13. to pause at verses that promise Allah's favour, to long for Allah Most High and ask of His bounty; and at verses that warn of His punishment to ask Him to save one from it;
14. to pause at the accounts of bygone peoples and individuals to heed and benefit from their example;
15. to find out the meanings of the Qur'an's unusual lexical usages;
16. to give each letter its due so as to clearly and fuLly pronounce every word, for each letter counts as ten good deeds;
17. whenever one finishes reciting, to attest to the veracity of ones's Lord, and that His messenger (Allah bless him and grant him peace) has delivered his message, and to testify to this, saying: "Our Lord, You have spoken the truth, Your messengers have delivered their tidings, and bear witness to this. O Allah, make us of those who bear witness to the truth and who act with justice": after which one supplicates Allah with prayers.
18. not to select certain verses from each surah to recite, but rather the recite the whole surah;
19. if one puts down the Qur'an, not to leave it open;
20. not to place other books upon the Qur'an, which should always be higher than all other books, whether they are books of Sacred Knowledge or something else;
21. to place the Qur'an on one's lap when reading; or on something in front of one, not on the floor;
22. not to wipe it from a slate with spittle, but rather wash it off with water; and if one washes it off with water, to avoid putting the water where there are unclean substances (najasa) or where people walk. Such water has its own inviolability, and there were those of the early Muslims before us who used water that washed away Qur'an to effect cures.
23. not to use sheets upon which it has been written as bookcovers, which is extremely rude, but rather to erase the Qur'an from them with water;
24. not to let a day go by without looking at least once at the pages of the Qur'an;
25. to give one's eyes their share of looking at it, for the eyes lead to the soul (nafs), whereas there is a veil between the breast and the soul, and the Qur'an is in the breast.
26. not to trivially quote the Qur'an at the occurrence of everyday events, as by saying, for example, when someone comes, "You have come hither according to a decree, O Moses" [Qur'an 69:24], 
27. or,  "Eat and drink heartily for what you have done aforetimes, in days gone by" [Qur'an 69:24], when food is brought out, and so forth; 
28. not to recite it to songs tunes like those of the corrupt, or with the tremulous tones of Christians or the plaintiveness of monkery, all of which is misguidance; 
29. when writing the Qur'an to do so in a clear, elegant hand;
30. not to recite it out aloud over another's reciting of it, so as to spoil it for him or make him resent what he hears, making it as if it were some kind of competition;
31. not to recite it in marketplaces, places of clamour and frivolity, or where fools gather;
32. not to use the Qur'an as pillow, or lean upon it;
33. not to toss it when one wants to hand it to another;
34. not to miniaturize the Qur'an, mix into it what is not of it, or mingle this worldly adornment with it by embellishing or writing it with gold;
35. not to write it on the ground or on walls, as is done in some new mosques;
36. not to write an amulet with it and enter the lavatory, unless it is encased in leather, silver, or other, for then it is as if kept in the heart;
37. if one writes it and then drinks it (for cure or other purpose), one should say the Basmala at every breath and make a noble and worthy intention, for Allah only gives to one according to one's intention;
38. and if one finishes reciting the entire Qur'an, to begin it anew, that it may not resemble something that has been abandoned.

thanks to: Mas'ud Ahmed Khan