  We show that many-body localization (MBL) effects can be observed in a finite
chain of exchange-coupled spin qubits in the presence of both exchange and
magnetic noise, a system that has been experimentally realized in
semiconductors and is a potential solid-state quantum computing platform. In
addition to established measures of MBL, the level spacing ratio and the
entanglement entropy, we propose another quantity, the spin-spin correlation
function, that can be measured experimentally and is particularly well-suited
to experiments in semiconductor-based electron spin qubit systems. We show
that, in cases that the established measures detect as delocalized "phases",
the spin-spin correlation functions retain no memory of the system's initial
state (i.e., the long-time value deviates significantly from the initial
value), but that they do retain memory in cases that the established measures
detect as localized "phases". We also discover an interesting counterintuitive
result that there is no clear tendency towards localization with increasing
charge noise in small systems ($3$--$10$ spins). The proposed experiments
should be feasible in the existing semiconductor spin qubit systems.
