Why Doesn't Compressive Sensing Work for Any Signal? 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?
You can measure and reconstruct such a 1-sparse signal as you describe in your question. The crucial misunderstanding here is, as @MBaz points out, that you have to know the basis $\Psi$.

You can take measurements in a universal manner without knowing the dictionary $\Psi$ beforehand, if you choose the measurement matrix $\Phi$ as a sufficiently random matrix - could be a random Gaussian matrix. What universality gives you is the ability to reconstruct $x$ again with good accuracy almost no matter what $\Psi$ turns out to be. You can take such universal measurements at the encoder without knowing $\Psi$, but you still have to know some $\Psi$, that admits a sparse representation of $x$, (and $\Phi$) at the decoder.