  We describe searches for decays to two-body charmless final states eta' eta,
eta' pi0 and eta pi0 of B0 mesons produced in e+e- annihilation. The data,
collected with the BaBar detector at the Stanford Linear Accelerator Center,
represent 232 million produced B-anti-B pairs. The results for branching
fractions are, in units of 10^-6 (upper limits at 90% C.L.): BF(B0 -> eta' eta)
= 0.2 +0.7 -0.5 +/- 0.4 (<1.7), BF(B0 -> eta pi0) = 0.6 +0.5 -0.4 +/- 0.1
(<1.3), and BF(B0 -> eta' pi0) = 0.8 +0.8 -0.6 +/- 0.1 (<2.1). The first error
quoted is statistical and the second systematic.
