How come most Lisps and Schemes are dynamically typed? Does static typing not mix with some of their common features?
Typing and s-expressions can be made to work together, see typed scheme.
Partly it is a historical coincidence that s-expression languages are dynamically typed. These languages tend to rely more heavily on macros, and the ease of parsing and pattern-matching on s-expressions makes macro processing much easier. Most research on sophisticated macros happens in s-expression languages.
Typed Hygienic Macros are hard.
When Lisp was invented in the years from 1958 to 1960 it introduced a lot of features both as a language and an implementation (garbage collection, a self-hosting compiler, ...). Some features were inherited (with some improvements) from other languages (list processing, ...). The language implemented computation with functions. The s-expressions were more an implementation detail ( at that time ), than a language feature. A type system was not part of the language. Using the language in an interactive way was also an early implementation feature.
The useful type systems for functional languages were not yet invented at that time. Still until today it is also relatively difficult to use statically typed languages in an interactive way. There are many implementations of statically typed languages which also provide some interactive interface - but mostly they don't offer the same level of support of interactive use as a typical Lisp system. Programming in an interactive Lisp system means that many things can be changed on the fly and it could be problematic if type changes had to be propagated through whole programs and data in such an interactive Lisp system. note that Some Schemers have a different view about these things. R6RS is mostly a batch language generally not that much in the spirit of Lisp...
The functional languages that got invented later with static type systems then also got a non-s-expression syntax - they did not offer support for macros or related features. later some of these languages/implementations used a preprocessor for syntactic extensions.
Static typing is lexical, it means that all information about types can be inferred from reading source code without evaluating any expressions or computing any things, conditionals being most important here. A statically typed language is designed so that this can happen, a better term would be 'lexically typed', as in, a compiler can prove from reading the source alone that no type errors will occur.
In the case of lisp, this is awkwardly different because lisp's source code itself is not static, lisp is homo-iconic, it uses data as code and can to some extend dynamically edit its own running source.
Lisp was the first dynamically typed language, and probably for this reason, program code itself is no longer lexical in Lisp.
Edit: a far more powerful reason, in the case of static typing you'd have to type lists. You can either have extremely complex types for each lists which account for all elements, of demand that each element has the same type and type it as a list of that. The former option will produce hell with lists of lists. The latter option demands that source code only contains the same type for each datum, this means that you can't even build expressions as a list is anyhow a different type than an integer.
So I dare say that it is completely and utterly infeasible to realize.