This may sound like a obvious statement but much of the time when we are singing a long note (hopefully usually a vowel sound) we have a tendency to move our lips, or our tongue, or our face to help shape the note.
Whilst this may seem like an appropriate idea to develop the sound many times it makes the sound feel more unstable. The stiller your mouth is on long notes the more clear it usually sounds.
This may seem slightly unusual as much of the time we can feel that more of everything is required for using our mouth with singing and for consonants certainly more punch and therefore tongue, teeth and face action, is often required, but the more you can let the vowels sing out, and not move your face shape, you may find that your sound becomes more stable.
This is especially true when working on songs with regional accents as the accent can only tend to be given within a vowel (a 'T' or a 'B' tends to be constant as they are plosives), therefore it is helpful to just be aware of how much you are moving within the vowel to gain the accent required. You may find it is more helpful to wait until towards the end of the note you are singing before hinting towards the accent.
The vowel is the thing we 'sing' as plosives can't really have a note and the stiller your mouth can be whilst those vowels are taking place you may find a cleaner, clearer sound.
Still Lips : Singing Tips
