Singing is made up of several sets of skills.
The primary essential skill is that of musicianship. The strength of this or the weakness of it will always show up (often unbeknownst to the singer) and will be obvious to those educated in music. Education simply means, in this sense, that the singer hears music and his/herself knows what was heard, indentifies it, relates to it, understands it, could re-sing it or even write it down AND this has occurred from having somehow learned these skills. Inadequate musicianship always results in an inadequate singer. This is nor a fatal flaw, however. This is something which can be improved and doesn't necessarily have to take a long time.
There are several skills making up musicianship, some of which were mentioned above. Most music majors in college have a major instrument and a minor instrument. Most singing majors major in voice and minor in piano. Knowing piano is very useful in learning chords, harmony, and music theory. Voice majors take music theory. I teach music theory (I prefer the term harmonic technique because it is more precise and clear). Rather than a broad and general approach to the subject, I teach it from the perspective of a singer. This relationship from voice to musicianship builds more quickly due to the application of whatever musicianship initially exists being built to the expansion of knowledge and skill, as it directly relates to singing.