By the way, these examples are where Mathematica has issue:
InverseLaplaceTransform[1/(s^\[Alpha] - a), s, t] == t^(\[Alpha] - 1)*MittagLefflerE[\[Alpha], \[Alpha], a*t^\[Alpha]]
InverseLaplaceTransform[s^\[Alpha]/(s*(s^\[Alpha] + a)), s, t] == MittagLefflerE[\[Alpha], -a*t^\[Alpha]]
InverseLaplaceTransform[a/(s*(s^\[Alpha] + a)), s, t] == 1 - MittagLefflerE[\[Alpha], -a*t^\[Alpha]]
InverseLaplaceTransform[s^(\[Alpha] - \[Beta])/(s^\[Alpha] - a), s, t] == t^(\[Beta] - 1)*MittagLefflerE[\[Alpha], \[Beta], a*t^\[Alpha]]
and should also solve:
LaplaceTransform[t*f[t], t, s] == -D[LaplaceTransform[f[t], t, s], s]
LaplaceTransform[t*f'[t], t, s] == -LaplaceTransform[f[t], t, s] - s*D[LaplaceTransform[f[t], t, s], s]
...
LaplaceTransform[t^2*f[t], t, s] == D[LaplaceTransform[f[t], t, s], {s, 2}]
LaplaceTransform[t^2*f'[t], t, s] == 2*D[LaplaceTransform[f[t], t, s], s] + s*D[LaplaceTransform[f[t], t, s], {s, 2}]
...