Inplace sorting of variable length records
If we're just dealing with packed variable-length records, swapping two adjacent elements is not difficult, so bubble sort will work.   

There are some techniques for creating space while sorting to use for auxiliary data structures, based on the fact that (partially) sorted data has less entropy than unsorted.  See Radix Sorting With No Extra Space by Franceschini, Muthukrishnan, and Patrascu for examples.  They don't deal with variable length records specifically, but some of the ideas there might be useful.