First, we must limit ourselves to talking about instrumental training. Not singing lessons. This is not because singing couldn’t potentially improve speech (most evidence suggests it does) but because focusing on singing confuses our two interests (speech and music). Singing training involves heavy overlap with speech by virtue of the verbal content. So any impact of singing on speech processing could be partly explained by ‘within domain’ plasticity – and we are interested here in ’cross domain’ plasticity.
The best type of evidence for the effect of music on speech is longitudinal (carried out over a long period – think months) and experimental. That means randomly assigning people to a music training vs. an active control activity (e.g. painting). If testing children then you also need a passive control group (one that takes no extra lessons) to rule out the effects of simple maturity on the brain.