Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $1$-unconditional basis, it can also be viewed as a Banach lattice. Therefore Banach lattices need not contain isomorphic copy of any of the classical spaces $\ell_p$ or $c_0$.
What about in the case of a Banach lattice which is not discrete? Is there some positive result about the classical spaces which must be subspaces of a Banach lattice which is not discrete?