1. The invariant subspace problem for Banach spaces was solved in the negative for Banach spaces by Per Enflo and counterexamples for many classical spaces were constructed by Charles Reid.  The problem is open for reflexive Banach spaces.  On the other hand, S. Argyros and R. Haydon recently constructed a Banach space $X$ s.t. $X^*$ is isomorphic to $\ell_1$ and every bounded linear operator on $X$ is the sum of a scalar times the identity plus a compact operator, hence the invariant subspace problem has a positive solution on $X$.

2. The invariant subspace problem has spurred quite a lot of interesting mathematics. Usually when a positive result is proved, much more comes out, such as a functional calculus for operators.  See, e.g., recent papers by my colleague C. Pearcy and his collaborators.

3. In cases where the ISP has a positive solution for a class of operators, there may be a structure theory for the operators. There is, for example, J. Ringrose's classical structure theorem for compact operators on a Banach space. This is a beautiful and useful theorem, which, BTW, I am using currently with T. Figiel and A. Szankowski to relate the Lidskii trace formula to the J. Erdos theorem in Banach spaces. 

4. Why is the twin prime conjecture interesting?