  Cosmological models of the early or late universe exhibit (quasi) de Sitter
space-times with different stability properties. Considering models derived
from string theory, the swampland program does not provide for now a definite
characterisation of this stability. In this work we focus on de Sitter
solutions of 10d type II supergravities, candidates for classical de Sitter
string backgrounds: surprisingly, all known examples are unstable with $\eta_V
< -1$. We aim at proving the existence of such a systematic tachyon, and
getting formally a bound on the value of $\eta_V$. To that end, we develop
three methods, giving us various sufficient conditions for having a tachyon
upon assumptions, in analogy with de Sitter no-go theorems. Our analysis
eventually indicates the existence of variety of different tachyons, and
related bounds on $\eta_V$. We use this knowledge to find 10 new de Sitter
solutions of type IIB supergravity, that have tachyons of a different kind,
higher $\eta_V$ values and new 6d geometries. One solution even appears to be
stable, with however non-compact extra dimensions.
