The question of music is a complex one and there are valid differences of opinion surrounding it. As a quick introductory statement, we as Muslims must realise that understand the Fiqh (Legal Ruling) of any matter is not as simple as quoting a Qur'anic verse or pointing to a Hadeeth. In fact, the Companions, and definitely the two generations following them (Tabi'een and Tabi' Tabi'een) refrained from quoting from the Sunnah or the Qur'an without proper analysis of many many issues, e.g., linguistic proofs, context of revelation as well as a narration of the Prophet (upon whom be peace and blessings of Allah), absolute and qualified narrations, universality and particularity of verses and narrations, etc. These are deeply developed sciences. The points below are the summaries of rulings based on such a thorough understanding of the Shari'ah, rather than those who simply claim to quote a verse or a Hadith as proof.
Further, it is also important to note that music and singing are essentially two different things, not mutually exlusive but certainly not dependent of each other. In other words, one does not have to have music to sing, but at the same time they are both musical matters.
In regards to the question itself, the matter can be broken down into the following categories of assessment, each holding its own ruling and differences of opinion.
We will start from those matters unanimously agreed upon to be prohibited.
1. The kinds of singing which are unequivocally prohibited are those which contain the celebration of the material world and includes sexual connotation and that which is also inappropriate in speaking, e.g., swearing, sexually expletive language, and the like. In Fiqh this is usually referred to as Tarab. Almost all Hollywood (and Bollywood...) songs will fit in here.
2. The kinds of singing which are also unequivocally prohibited, are those that remove a person away from the worship and appropriate presence with Allah, e.g., leading a person to be involved with cross-gender mixing, lazing around (rather than taking a short break to relax from exhaustion. In such cases, as we will see below, there is permission to listen to musical matters that glorify Allah and praise the Prophet), ignoring one's rights and responsibilities, and the like.
3. Those kinds of singing which are unequivocally permitted are those which glorify Allah and praise the Prophet. Much can be said about this, but it will be a whole book! So in summary, this is usually taken to be permissible as a respite rather than the norm of "iPod in the ear 24/7". It is also permissible to have such appropriate singing of happiness (rather than directly about Allah and His Messenger) that soothe the heart in a Halal manner for festivals and weddings. These kinds of singing are usually referred to as inshad and sama'.
4. Those kinds of singing which are general, and are neither prohibited nor specifically about goodness and happiness (but may be e.g., about politics or environment, etc), the majority of Scholars hold that it is permissible only in its context rather than a habit to listen and enjoy. If it is habituated and leads an individual to ignore their rights and responsibilities, then it is not permitted. Almost all Hollywood and Bollywood songs fit in here.
Now to music:
1. The kind of music referred to as malahi are abhorred and prohibited. This kind of music is simply for entertainment for dance, frivolous enjoyment and the like.
2. Those kinds of music from the duff and similar drums (under the category of ma'azif) are unequivocally permissible. Of course, the songs attached to it must be permissible as outlined above. Further the use of the drums are unequivocally permissible for festivals and weddings and joyous times.
3. The critical difference of opinion is regarding different kinds of musical instruments; it is incorrect to suggest that all musical instruments are held to be prohibited in Islam. The fact is that the major prohibition of music is because at the time of the Prophet, they very often were used in the context of malahi - (1) above, hence easily claimed that it is prohibited in all circumstances. But this is not the case. As such (and the same for singing with or without music):
i) If the music is part of glorifying Allah and His Messenger, this is held to be permissible by the majority of Scholars;
ii) If the music is for entertainment and pastime, it is prohibited;
iii) If the music is for soothing and serenity of the mind which helps one to remember Allah, as long as it does not remove one from obeying Allah or one's rights and responsibilities, there is a significant difference of opinion, and is best to minimise such involvement if not avoided. If you saw someone listening to such a musical matter, do not place judgment on them nor force your opinion on others regarding this category.
In conclusion, avoid all singing, music, gatherings of entertainment, except for gatherings of melodious remembrance of Allah and His Messenger. And, be cautious if singing and music which are of spiritually happiness (not material) and also leads one to remember Allah and only listen sparingly if one could not avoid it at all.
Insha Allah that gives you a practical set of guidelines and the scope of permissibility as well as definite prohibitions and as such help you to live your life better in the presence of Allah Ta'ala.