In classical singing it is the vowels that should be held, and you should "sing through the consonants" in order to maintain a smooth tone.
In any singing there has to be a compromise between beautiful sounds and comprehensibility. That compromise comes at different places for different styles.
For the majority of folk songs it is the text that is the most important thing - and so the compromise point will reflect that fact. In such cases there may well be instances where it is more appropriate and expressive to hold the consonant. Beauty of tone can be sacrificed in favour of telling the story - and this effect can be a big part of the flavour and emotion of the song.
The classical style is more weighted toward formal tonal beauty, and well written songs will allow for the fact that vowels must be modified etc - and will ensure that the important words and concepts of the song are carried across in the places where the text is more accessible - and the less audible text will be understandable by context.
In both styles the issue is one of communication, and the transfer of information. So it is horses for course as to how you chose to do it in any one song/situation.
Singing is not equivalent to speaking - ie it is not just speech on different pitches. Vowels are formed by the relationship of certain resonant peaks, and this is why some vowels are easy on a particular note (ie the singer is producing the required resonances)whilst others are very difficult on the same note (ie the required resonances are not there). So you will find that some vowels are easier to hold than others - which (in nonclassical singign) might be a good reason for holding on a consonant if that prooves easier.

Kris