Why do people who stutter have less difficulty singing than speaking? From what I have read, stutterers tend to have much less trouble singing than speaking.  Do we know why this is the case?
Short answer
Singing increases the duration of voiced intervals in stutterers.

Background
Singing is an example of one of the most effective methods to decrease stuttering* (Stager, 2003). It is a so-called fluency-increasing (FI) condition in stutterers and reduces stuttering by more than 90%. Some of the few, subtle acoustic differences between song and speech are the more isochronous rhythm and greater fundamental frequency stability within each syllable in songs (Tierney et al., 2012). Other effective fluency-evoking conditions are speech pacing (e.g., with a metronome), chorus speaking, speech slowing, and whispering (Andrews et al., 1982). 

The main effects of FI conditions (including singing) are an increase in phonation duration and speech slowing. The reduction of stuttering seems most directly related to phonation. Phonation is the production of vocal sounds, i.e., vowels. Specifically, the frequency of very short (in the range of 30–150 ms) phonated intervals is decreased, and the longer duration phonated intervals are increased when stutterers are singing. Indeed, it has been suggested that the most effective FI conditions, and perhaps therefore the most effective treatments for stuttering, might share the common property of reducing the number or proportion of short intervals of phonation, or of quick transitions between phonated and nonphonated speech production, and/or might be associated with an increased number or proportion of longer, slower, or more consistent phonation. Among the best supported treatment approaches for stuttering are those that emphasize increased, prolonged, or continuous phonation and/or slower initiation of speech (referred to as prolonged speech treatment programs) (Ingham, 2012).

Most fluency-evoking conditions, however, have only temporary effects. Moreover, normal fluency has been defined for stuttering in terms of four variables or characteristics: the absence of stuttering events, speech rate within a normal
range, normal overall naturalness of speech, and normal levels of physical and cognitive effort on the part of the speaker. Of the various methods used, only chorus speaking meets these four demands (Ingham, 2012). 

References
- Andrews et al., J Speech Hear Res (1982); 25(2): 208-16
- Ingham et al., J Comm Disorders (2012); 45: 198–211
- Stager et al., J Fluency Disorders (2003); 28: 319–36
- Tierney et al., Cereb Cortex (2012); 23(2):249-54  

Note
* Being a stotterer myself, I wish to emphasize that when the speech therapist at primary school instructed me to sing my presentation to my class mates to reduce my stuttering I chose the less embarrassing alternative, namely stammering unintelligibly for half an hour about the planets in the solar system instead of singing about them. One can quickly envision the utter non-practicality of the singing approach in everyday life. The effect of FI conditions is scientifically interesting, and they are useful tools for speech therapists, but they are not practical treatments, unfortunately.