  We construct compact and high accuracy Reduced Basis (RB) representations of
single and multiple quasinormal modes (QNMs). The RB method determines a
hierarchical and relatively small set of the most relevant waveforms. We find
that the exponential convergence of the method allows for a dramatic
compression of template banks used for ringdown searches. Compressing a catalog
with a minimal match $\MMm=0.99$, we find that the selected RB waveforms are
able to represent {\em any} QNM, including those not in the original bank, with
extremely high accuracy, typically less than $10^{-13}$. We then extend our
studies to two-mode QNMs. Inclusion of a second mode is expected to help with
detection, and might make it possible to infer details of the progenitor of the
final black hole. We find that the number of RB waveforms needed to represent
any two-mode ringdown waveform with the above high accuracy is {\em smaller}
than the number of metric-based, one-mode templates with $\MMm=0.99$. For
unconstrained two-modes, which would allow for consistency tests of General
Relativity, our high accuracy RB has around $10^4$ {\em fewer} waveforms than
the number of metric-based templates for $\MMm=0.99$. The number of RB elements
grows only linearly with the number of multipole modes versus exponentially
with the standard approach, resulting in very compact representations even for
many multiple modes. The results of this paper open the possibility of searches
of multi-mode ringdown gravitational waves.
