What kinds of programming pitfalls modern languages are able to find?
Substructural types ( http://en.wikipedia.org/wiki/Substructural_type_system ) can be used to provide some version of each of those. As a summary, while normal types control how something may be used, substructural types further control when or how often it may be used.


Affine types can be used to construct languages capable of expressing all, and only, polynomial-time functions. http://www.cs.cmu.edu/~fp/courses/15816-s12/misc/aehlig02tocl.pdf
Linear types can enforce that memory is accessible when variables go out of scope, enabling a kind of "static garbage collection." Similarly, ordered types can be used to require that variables are used in a LIFO order, allowing them to be stack-allocated. Regions ( http://en.wikipedia.org/wiki/Region-based_memory_management ) are also useful here. The first chapter of Advanced Topics in Types and Programming Languages provides a nice introduction to using linear and ordered types for this purpose.
Affine types also enable typestate-oriented programming ( http://www.cs.cmu.edu/~aldrich/papers/onward2009-state.pdf ). Typestate is used to enforce certain orderings in APIs (e.g.: can't read from a closed file), but can also be used to enforce safety properties. Requiring a block that handles some error can be encoded as requiring a function that transitions some object from an error state to a recovery state.