Some people call the Qur'an, "The Qur'an-i-Shareef" as a mark of respect. What follows are the thoughts of Imam Muhammad ibn Ahmad Qurtubi who was a great Qur'an Scholar and renowned exegisate of the Qur'an, taken from the Shari'ah manual
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 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:
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 interuppt 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],
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;
27 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;
28 when writing the Qur'an to do so in a clear, elegant hand;
29 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;
30 not to recite it in marketplaces, places of clamour and frivolity,
or where fools gather;
31 not to use the Qur'an as pillow, or lean upon it;
32 not to toss it when one wants to hand it to another;
33 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;
34 not to write it on the ground or on walls, as is done in some new
mosques;
35 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;
36 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;
37 and if one finishes reciting the entire Qur'an, to begin it anew,
that it may not resemble something that has been abandoned.
We now see people with the Qur'an-i-Shareef on their laps, and at the foot of their beds (in bookcases etc). It is important that one truly show respect the Qur'an as a Word from God. i.e. wudh'u (ablutions) before touching it, or reading it, never have your feet higher or pointing at the Holy Book, do not place any object on top of it and so on. Still, it is good to see that people are at least reading the Qur'an. It is far better to do this than merely place it in a place of honour and forget about it.
One must not treat the Holy Qur'an as just as an ordinary book, as it is the Word of God Himself, may He be Glorified and Exalted. In this conduct we must take our example from those who are our predecessors. We see that they showed the Qur'an the utmost respect,
The four schools are in agreement that one has to be in wudhu to handle the Qur'an (The Shafi'i's allow handling the Qur'an without wudhu for purposes of study but stress the importance of wudhu for it).