This is something I have always wondered.
Is singing a matter of talent, that is, only people born with the 'proper instrument' can sing properly, or is it a matter of practice?
I know that practice alone can only take you so far, but how far is that?
Both, but talent and practice tend to refer to very nearly the same thing (It's hard to separate innate musical ability from ability conditioned from dedicated practice). Let's clarify your question as "Genetics or practice".
Practice will take you into the realm of professional singing. It's not enough to get you big solos, but it is enough to get singers into professional choirs and the like. Of course some people will pick up singing easier than others, but I liken this effect to how some people pick up calculus easier than others; a dedicated student with real interest in either subject will be able to learn.
This leads into a point that Marian and luser have made before; a lack of interest is not a lack of potential; it's just that they don't want to learn. On the flip side, it is entirely possible that some people did not develop a basic understanding of music when they were children, and so are unable to do basic exercises like matching pitch. These people can still learn, although it takes a very long time to develop their musical ear.
It's also worth noting that in the sense I'm using it, vocal instruction is very comprehensive, from dictating how to breathe to where exactly in the mouth your tongue moves to make a consonant or vowel. Things like voice placement (nasal versus almost swallowed) and color (bright versus dark) can take years to get alone, and they're skills required for choral singing. With this level of minutiae, it's no wonder that people give up and go home; practicing is not always fun.
Compare that to genetics. Genetics will influence your tone (by the shape of your pharynx) and, possibly, your ability to learn. It's almost not fair, because regardless of your technique, you need an astounding tone to be able to get the big high-profile roles. The converse is not always true, though; those singers gifted with a good voice are not putting any less effort into their craft than others with a lesser tone.
So, the answer: Let's consider a scene from an opera, with a prima donna and the chorus. Practice alone will get you into the chorus. Genetics and practice will get you into the prima donna role. Genetics without practice will get you into the audience! :P
Everyone can sing, but some people can sing better than others. This is true apart from the very tiny number of people who have a condition called amusia. People with amusia derive absolutely no pleasure from music, and can even find it very uncomfortable. They have no understanding of why people move to music, tap their feet, hum, etc, and can often feel socially isolated because of it. Fortunately, in a world with music in every shop, this is a very rare condition.
"But I'm tone deaf". This is a way of people saying that they can't carry a tune, or can't say if a pitch is high or low, and it's about lack of childhood training.
"I hate singing". This usually means, actually I like singing in the shower when I'm alone and no one can hear me, but I hate the idea of performing. This is completely understandable, especially because most people have considerable awareness of not being on the notes.
"I can't sing". Lots of people who say this have had really horrible experiences of being rejected from the school choir, told to "sing more quietly - you're out of tune", moved to the back of the choir for the same reason, and so on. I've met school teachers with lovely voices, frightened to use them, due to a combination of lack of training (how can you train someone if you don't let them in to the group?) and inadvertently hurtful treatment.
There are lots of things you can do to help children and adults find their singing voice, even those who start off droning on one note. Recently I helped a child move from being a "droner" to a singer in ten minutes in a group activity. That wasn't just down to my skills - he had it in him all the time, although his class teacher said it was a little miracle.
The other part of all this is there are natural limits to what each person can do, which depend on the vocal chords and lung capacity nature gave you. You can still improve your tone, range and expression enormously through practice, but there'll always be professional singers gifted with both exceptional musicianship and vocal apparatus.
The noted right answer is not fully correct. All genetics does is give a person a base starting point. This starting point in a voice is either one that tends to have mass appeal or one that has lesser appeal. You can by-pass genetics to end up with a voice that has mass appeal. To do this you need a lot of practice and an understanding of how you attain your goals. We prove what is being said here by noting that people can modifiy their voice to sound like someone else. However, doing so means you're a copy. This is where the trouble comes in. People born with great tone just have to do less work for listeners to say "wow I like the way his or her voice sounds". The true question is "where do you take your voice so that is original and has mass appeal. That's the magic. That's were higher powers have stepped in to give gifts by allowing the choosen to be born with the magic genetics - the so called default settings that people tend to love.
If you want to have a unique yet magic voice here is what you do: Once you have the base singing talent via practice or again good genetics - you record a record. Then by a process of repeated modifying of your voice/tone and recording that sample, you end up with an infinite amount of versions of the recording. Now, just get a large group of people together to rate all the version. Eventuall they will come up with an iteration that sounds the best and that most people tend to like. Simple ;-)
Since it would be unethical to actually investigate this by direct experiment (since all "nature v. nurture" experiments would require raising a child in total isolation), we only ever apply these labels and judgements after the fact.
So the distinction between "talent" and "practice" seems to me to rest on the emotional component. One with "talent" derives such joy from the exercise of music that "it doesn't feel like practice." But to achieve mastery, even the talented have put in the time.
I have even less information on the distribution of crucial physical defects of the vocal apparatus in the human population, but I'd hazard a guess that most people could sing well if they "truly want to" (there's that emotional component again). But again, we can only apply this label after-the-fact. Anyone who can't sing well, but "have the chords" must not "truly want to", otherwise they would naturally have performed the requisite practice.
So, in sumary, i dunno.
My wife once day-cared a little girl, less than two years old. She would sing spontaneously every once in a while, mostly children songs she'd heard before. She hit every note dead on. Completely effortlessly and supremely confident she nailed every single note every time although she was still too young to get the words out. Talent does exist and it certainly helps.