Compression type that can be searched
Compressed self-indexes such as the FM Index allow arbitrary substring searches in near entropy-compressed space.  These are essentially compressed suffix arrays or suffix trees, which have a lot of literature.  

Basic substring search can be o(k) or o(k log n) in time for length k, depending on what data structures are chosen (different types of rank/select data structures).  There are a range of issues that arise depending on whether one wants simple boolean containment predicates, the offset of each occurrence, or more complicated suffix tree operations; the former can be done in less space and time than the latter.  

There's also an entire book about searching and selective decompression of strings: "Compressed Data Structures for Strings: On Searching and Extracting Strings" by Rossano Venturini, published 2014 Springer Science &amp; Business Media.