  Tsirelson's space $\mathcal{T}$ made its appearance in Banach space theory in
1974 soon to become one of the most significant counterexamples in the theory.
Its structure broke the ideal pattern that analysts had conceived for a generic
Banach space, thus giving rise to the era of pathological examples. Since then,
many authors have contributed to the study of different aspects of this special
space with an eye on better understanding its idiosyncrasies. In this paper we
are concerned with the greedy-type basis structure of $\mathcal{T}$, a subject
that had not been previously explored in the literature. More specifically, we
show that Tsirelson's space and its convexifications $\mathcal{T}^{(p)}$ for
$0<p<\infty$ have uncountably many non-equivalent greedy bases. We also
investigate the conditional basis structure of spaces $\mathcal{T}^{(p)}$ in
the range of $0<p<\infty$ and prove that they have uncountably many
non-equivalent conditional almost greedy bases.
