Resummation and power-corrections play a crucial role in the phenomenology of event-shape variables like the thrust T. Previous investigations showed that the perturbative contribution to the average thrust is dominated by gluons of small invariant mass, of the order of 10 of Q, where Q is the center-of-mass energy. The effect of soft gluons is also important, leading to a non-perturbative 1/Q correction. These conclusions are based on renormalon analysis in the single dressed gluon (SDG) approximation. Here we analyze higher moments of the thrust distribution using a similar technique. We find that the characteristic gluon invariant mass contributing to ⟨(1−T)^m⟩ increases with m. Yet, for m = 2 this scale is quite low, around 27 of Q, and therefore renormalon resummation is still very important. On the other hand, the power-correction to ⟨(1−T)²⟩ from a single soft gluon emission is found to be highly suppressed: it scales as 1/Q³. In practice, ⟨(1−T)²⟩ and higher moments depend also on soft gluon emission from configurations of three hard partons, which may lead to α_s(Q²)/Q power-corrections. This issue is yet to be investigated.