  We explore the multi-superthreads and supersheets solutions of
six-dimensional N=1 supergravity coupled to a tensor multiplet. The solutions
carry D1-D5-P charges, but no Kaluza-Klein monopole. We lay down the formalism
to construct multiple supersheets with arbitrary and independent profiles. The
solution is by construction free of Dirac strings in contrast to the
five-dimensional construction where one has to separately solve integrability
conditions. We explore this formalism to construct supersheets that fluctuate
in both directions allowing a more general choice of profiles. These new
solutions are genuinely six-dimensional, singular, fluctuating BPS solutions
and by analyzing them we expect to learn more about the conjectured
superstrata. We also derive the conditions under which different supersheets
can touch, or even intersect through each other.
