There will be no spitting in Paradise.
Narrated Abu Huraira: Allah's Apostle said, "The first group (of people) who will enter Paradise will be (glittering) like the moon when it is full. They will not spit or blow their noses or relieve nature. Their utensils will be of gold and their combs of gold and silver; in their centers the aloe wood will be used, and their sweat will smell like musk. Everyone of them will have two wives; the marrow of the bones of the wives' legs will be seen through the flesh out of excessive beauty. They ( i.e. the people of Paradise) will neither have differences nor hatred amongst themselves; their hearts will be as if one heart and they will be glorifying Allah in the morning and in the evening." (Sahih al-Bukhari: volume 4, book 54, number 468)
Spitting to your left side is a way to be safe from a bad dream.
Narrated Abu Qatada: The Prophet said, "A good dream is from Allah, and a bad or evil dream is from Satan; so if anyone of you has a bad dream of which he gets afraid, he should spit on his left side and should seek Refuge with Allah from its evil, for then it will not harm him." (Sahih al-Bukhari: volume 4, book 54, number 513)
Spitting to your left three times stops the interference of Satan.
Uthman b. Abu al-'As reported that he came to Allah's Messenger (may peace be upon him) and said: Allah's Messenger, the Satan intervenes between me and my prayer and my reciting of the Qur'an and he confounds me. Thereupon Allah's Messenger (may peace be upon him) said: That is (the doing of the Satan) who is known as Khinzab, and when you perceive its effect, seek refuge with Allah from it and spit three times to your left. I did that and Allah dispelled that from me. (Sahih Muslim: book 026, number 5463)
Muhammad spat on people and objects to bless them.
Narrated Mahmud bin Ar-Rabi': I remember Allah's Apostle and also the mouthful of water which he took from a bucket in our house and ejected (on me). ... (Sahih Bukhari: volume 1, book 12, number 801)
... Urwa returned to his people and said, "O people! By Allah, I have been to the kings and to Caesar, Khosrau and An-Najashi, yet I have never seen any of them respected by his courtiers as much as Muhammad is respected by his companions. By Allah, if he spat, the spittle would fall in the hand of one of them (i.e. the Prophet's companions) who would rub it on his face and skin; if he ordered them, they would carry out his order immediately; if he performed ablution, they would struggle to take the remaining water ... (Sahih al-Bukhari: volume 3, book 50, number 891)
Narrated Jabir bin 'Abdullah: ... So I came (to my house) and Allah's Apostle too, came, proceeding before the people. When I came to my wife, she said, "May Allah do so-and-so to you." I said, "I have told the Prophet of what you said." Then she brought out to him (i.e. the Prophet) the dough, and he spat in it and invoked for Allah's Blessings in it. Then he proceeded towards our earthenware meat-pot and spat in it and invoked for Allah's Blessings in it. (Sahih al-Bukhari: volume 5, book 59, number 428)
Muhammad spat in Ali's eyes to bless them, (Sahih al-Bukhari: volume 4, book 52, number 192)
Muhammad spat in a well to bless it, (Sahih al-Bukhari: volume 5, book 59, number 472)
Related to Muhammad spitting on people is the reverence that was given to his used ablution water.
Narrated Abu Juhaifa: I saw Allah's Apostle in a red leather tent and I saw Bilal taking the remaining water with which the Prophet had performed ablution. I saw the people taking the utilized water impatiently and whoever got some of it rubbed it on his body and those who could not get any took the moisture from the others' hands. ... (Sahih al-Bukhari: volume 1, book 8, number 373)
Related topic: RELICS