There is nothing wrong with the person who is initiating the greeting saying, “Salaamun ‘alaykum” or ‘Salaamun ‘alayk.” Allaah tells us that the greeting of the angels to the people of Paradise will be “Salaamun ‘alaykum” as He says (interpretation of the meaning): 

“And angels shall enter unto them from every gate (saying):

24. ‘Salaamun ‘Alaykum (peace be upon you) for you persevered in patience! Excellent indeed is the final home!’”

[al-Ra’d 13:23, 24]

“And those who kept their duty to their Lord (Al-Muttaqoon – the pious) will be led to Paradise in groups till when they reach it, and its gates will be opened (before their arrival for their reception) and its keepers will say: Salaamun ‘Alaykum (peace be upon you)! You have done well, so enter here to abide therein”

[al-Zumar 39:73] 

The salaam in this form is also mentioned in the verse where Allaah says (interpretation of the meaning): 

“Those whose lives the angels take while they are in a pious state (i.e. pure from all evil, and worshipping none but Allaah Alone) saying (to them): Salaamun ‘Alaykum (peace be on you) enter you Paradise, because of that (the good) which you used to do (in the world)”

[al-Nahl 16:32] 

“And when they hear Al‑Laghw (dirty, false, evil vain talk), they withdraw from it and say: ‘To us our deeds, and to you your deeds. Peace be to you (Salaamun ‘Alaykum). We seek not (the way of) the ignorant’”

[al-Qasas 28:55] 

“When those who believe in Our Ayaat (proofs, evidences, verses, lessons, signs, revelations, etc.) come to you, say: Salaamun ‘Alaykum (peace be on you); your Lord has written (prescribed) Mercy for Himself, so that if any of you does evil in ignorance, and thereafter repents and does righteous good deeds (by obeying Allaah), then surely, He is Oft‑Forgiving, Most Merciful”

[al-An’aam 6:54] 

Ibn Hibbaan narrated in his Saheeh (493) from Abu Hurayrah (may Allaah be pleased with him) that a man passed by the Messenger of Allaah (peace and blessings of Allaah be upon him) when he was in a gathering and said, “Salaamun ‘alaykum.” He (the Prophet (peace and blessings of Allaah be upon him)) said, “Ten hasanahs.” Another man passed by and said, “Salaamun ‘alaykum wa rahmat-Allaahi (Peace be upon you and the mercy of Allaah).” He said, “Twenty hasanahs.” Then another man passed by and said, “Salaamun ‘alaykum wa rahmat-Allaahi wa barakaatuhu (Peace be upon you and the mercy of Allaah and His blessings).” He said: “Thirty hasanahs.” A man got up and left the gathering and did not say salaam. The Prophet (peace and blessings of Allaah be upon him) said, “How soon your companion has forgotten. When one of you comes to a gathering and says salaam, if he wants to sit down then let him do so. Then when he wants to leave, let him say salaam, for the one is not more important than the other.” Classed as saheeh by al-Albaani in Saheeh al-Targheeb wa’l-Tarheeb, 2712. 

This and other evidence indicates that there is nothing wrong with saying salaam to a person by saying “Salaamun ‘alaykum.”  He will be rewarded for that, and he deserves a response to his salaam. 

The scholars differed as to which version is better, al-salaamu ‘alaykum or salaamun ‘alaykum, or whether they are both the same. 

Al-Mardaawi said in al-Insaaf (2/563): If a person greets a living person, then the correct view in our madhhab is that he has the choice between using the definite form (with al-) or the indefinite (without al-). And he said: this was stated by more than one. 

Then he mentioned a report from Imam Ahmad which says that the definite form is better than the indefinite, and he quoted Ibn ‘Aqeel as saying that the indefinite is better than the definite.  

Al-Nawawi said in al-Adhkaar (p. 356-358): 

Note that it is preferable for the Muslim to say “al-salaamu ‘alaykum wa rahmat-Allaahi wa barakaatuhu”, using the plural pronoun (-kum), even if he is greeting only one person. The respondent should say, “Wa ‘alaykum al-salaam wa rahmat-Allaahi wa barakaatuhu.” 

Our companions said: If the one who is initiating the greeting says, “Al-salaamu ‘alaykum,” then he has given the greeting. If he said “al-salaamu ‘alayk” or “salaamun ‘alayk” this also counts as the greeting. 

As for the response, the minimum is to say, ‘Wa ‘alayk al-salaam” or “Wa ‘alaykum al-salaam.” If the waw is omitted and one says, “ ‘Alaykum al-salaam” that is sufficient and is a response. 

If the person who is initiating the greeting says “Salaamun ‘alaykum” or “al-salaamu ‘alaykum,” in both cases the respondent may say, “Salaamun ‘alaykum”, or he may say, “al-salaamu ‘alaykum”. Allaah says (interpretation of the meaning): 

“When they came in to him and said: ‘Salaam, (peace be upon you)!’ He answered: ‘Salaam, (peace be upon you ),’”

[al-Dhaariyaat 51:25]

Imam Abu’l-Hasan al-Waahidi, one of our companions, said: You have the choice between using the definite form (al-salaam) or the indefinite (salaamun). 

I (al-Nawawi) say: But the alif and laam are preferable (i.e., saying the word salaam with the definite article al-). 

End quote. 

Secondly: 

What is makrooh is for the one who is initiating the greeting to say “ ‘Alayk al-salaam” or “ ‘Alaykum al-salaam,” because this is the greeting given to the dead as the Prophet (peace and blessings of Allaah be upon him) said. 

Abu Dawood (5209) and al-Tirmidhi (2722) narrated that Abu Jurayy al-Hujaymi (may Allaah be pleased with him) said: I came to the Prophet (peace and blessings of Allaah be upon him) and said: “ ‘Alayka’l-salaam (upon you be peace), O Messenger of Allaah.” He said, “Do not say ‘ ‘Alayka’l-salaam’, for ‘Alayka’l-salaam’ is the greeting of the dead.” This hadeeth was classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

What is meant by the Prophet’s words “‘Alayka’l-salaam’ is the greeting of the dead” is the fact that many poets and others greeted the dead in this manner. Moreover the Sunnah of the Prophet (peace and blessings of Allaah be upon him) is to greet the dead in the same manner as he greeted the living, by saying “Al-salaamu ‘alaykum.” 

Ibn al-Qayyim (may Allaah have mercy on him) said, explaining that: His greeting, when he was initiating the greeting, was to say, “Al-salaamu ‘alaykum wa rahmat-Allaah” He did not like the initiator to say “‘Alayka’l-salaam.” Abu’l-Jurayy al-Hujaymi said: I came to the Prophet (peace and blessings of Allaah be upon him) and said: “ ‘Alayka’l-salaam (upon you be peace), O Messenger of Allaah.” He said, “Do not say ‘ ‘Alayka’l-salaam’, for ‘Alayka’l-salaam’ is the greeting of the dead.” A saheeh hadeeth. 

This hadeeth confused some people, who thought that it contradicted what was narrated from the Prophet (peace and blessings of Allaah be upon him) about greeting the dead by saying, “Al-salaamu ‘alaykum,” saying the word salaam first. They thought that his words, “‘Alayka’l-salaam’ is the greeting of the dead” was describing what is prescribed, but that is a mistake which led them to think that there was a contradiction. Rather what is meant by the words “‘Alayka’l-salaam’ is the greeting of the dead” is a statement of something that used to happen, not something that was prescribed, namely that the poets and others used to greet the dead in this manner, and he did not like the greeting commonly used for the dead to become widespread.   

End quote from Zaad al-Ma’aad, 2/383 

Thirdly: 

The most complete form of greeting is to say “Al-salaamu ‘alaykum wa rahmat-Allaahi wa barakaatuhu (Peace be upon you and the mercy of Allaah and His blessings)” or “Salaamun ‘alaykum wa rahmat-Allaahi wa barakaatuhu”, because of the hadeeth of Ibn Hibbaan quoted above, and because of the hadeeth narrated by Abu Dawood (5195) and al-Tirmidhi (2689) from ‘Imraan ibn Husayn (may Allaah be pleased with him) who said: A man came to the Prophet (peace and blessings of Allaah be upon him) and said: “Al-salaamu ‘alaykum,” and he returned the greeting, then he sat down. The Prophet (peace and blessings of Allaah be upon him) said, “Ten.” Then another man came and said, “Al-salaamu ‘alaykum wa rahmat-Allaah,” and he returned the greeting, then he sat down. He said, “Twenty.” Then another man came and said Al-salaamu ‘alaykum wa rahmat-Allaahi wa barakaatuhu,” and he returned the greeting, then he sat down. He said, “Thirty.” Classed as saheeh by al-Albaani in Saheeh Abi Dawood.

 As for adding the words “wa maghfiratuhu (and His forgiveness)” or “wa ridwaanuhu (and His pleasure)”, these are not mentioned in any saheeh report from our Prophet (peace and blessings of Allaah be upon him), as was stated by Ibn al-Qayyim in Zaad al-Ma’aad, 2/381, and by al-Albaani in Da’eef Abi Dawood, 5196.  

And Allaah knows best.