My (probably naive) understanding of compressive sensing is that it is a technique that allows to efficiently reconstruct an $N$-dimensional signal $\boldsymbol x$, provided that it is sparse in some basis (without the need to know the sparsity basis), using a number $M\ll N$ of measurements.
More formally, if the signal can be written as $\boldsymbol x = \Psi\boldsymbol s$ with $\boldsymbol s$ $K$-sparse, and we are measuring in an incoherent $M$-dimensional measurement basis $\boldsymbol y = \Phi\boldsymbol x$ with $M\ll N$, then we can reconstruct the signal $\boldsymbol x$ using only $M$ measurements (provided $M\ge K$) (with the notation of Baraniuk 2007).
What I don't understand is why doesn't the above sparsity requirement apply to all possible signals. I mean, isn't any signal sparse in its own basis? Regardless of the choice of $\boldsymbol x$, I can always take as $\Psi$ any matrix whose first column is $\boldsymbol x$, and this will result in $\boldsymbol x$ being representable with the $1$-sparse vector $\boldsymbol s\equiv (1,0,0,...)^T$.
So why can't I efficiently reconstruct a signal which is $1$-sparse in a basis like the above? More generally, what are the requirements on the basis in which the signal has to be sparse for a CS approach to work?