Mathematica 11.3 is not aware of another useful Laplace transform
LaplaceTransform[t^(-a) MittagLefflerE[a, a, t^a], t, s]
which for 0 < a <= 1, Re[s] > 1 should be
(s^a - 1)^(-1)
as easily checked
s = 2; a = 1/2; Print[
NIntegrate[
Exp[-s t] t^(-a) MittagLefflerE[a, a, t^a], {t, 0, Infinity}], "=",
(s^a - 1)^(-1) // N]
(a similar remark was made above, in Mittag Leffler function Laplace transforms with Mathematica )
Note that in both cases we are talking about special cases of Mittag-Leffler functions, which should be easy to implement. They would allow however to implement very important "bread and butter" applications, like for example inverting Laplace transforms which are rational functions of s^a.