  We study the non-rigidity of Euclidean $t$-designs, namely we study when
Euclidean designs (in particular certain tight Euclidean designs) can be
deformed keeping the property of being Euclidean $t$-designs. We show that
certain tight Euclidean $t$-designs are non-rigid, and in fact satisfy a
stronger form of non-rigidity which we call strong non-rigidity. This shows
that there are plenty of non-isomorphic tight Euclidean $t$-designs for certain
parameters, which seems to have been unnoticed before. We also include the
complete classification of tight Euclidean $2$-designs.
