Things static type systems shouldn't have
If you've worked with any programming language, you have gripes about it. All design involves compromises, and all languages have warts. More interesting are the warts that are shared by many languages. Here's a short list of things that are extremely common in static type systems, that I think are almost always mistakes.
A populated bottom type
A non-denoteable type is a type which can't be written down in the programming language itself. Many languages with sophisticated static type systems require the use of non-denoteable types to typecheck the code. This almost guarantees that the programmer can't always reproduce the compiler's reasoning, or understand the resulting error messages.
Aggressively replacing unions with intersections
For some reason, there's some kind of prejudice against union types. Your typical compiler, upon encountering something internally that is naturally a union type, immediately attempts to replace it with the intersection of its supertypes. This doesn't make a whole lot of sense to me, since in these languages both unions and intersections are usually equally non-denoteable, and this operation involves throwing away a whole lot of useful information about the type.
Assigning references a single immutable type
A value has an immutable type. A reference to a value doesn't need to. Indeed, what the compiler knows (or rather could know) about the type of a reference changes depending upon precisely where we are in the code. But there's a strong tradition that each named reference has a single declared or inferred type everywhere it is used. This tradition even infects Ceylon to some extent (though this will change in future versions).
Primitive types or primitively-defined compound types
I hesitated to put this one on the list. There is an approach to building a static type system where we start with certain primitive values, layer tuple support over those types, layer records over tuples, and eventually arrive at generic abstract data types. I'm not talking about that kind of type system. What I'm talking about is where we start with a totally general way of defining abstract data types, and then stick some extra primitive types into the system because we find our supposedly totally general thing to be inadequate.
Overloading and implicit type conversions
Yes, yes, I know they seem convenient. But in fact the benefits provided by overloading or implicit type conversions are almost entirely trivial. They just don't increase the expressiveness of your type system. And they really screw up type inference, which is why you don't see these things in academic or research languages. Dynamic languages don't have anything like overloading or implicit type conversions, and nobody misses them. Because you don't need 'em. They're an abuse of the static type information.