NOTE: This information relates more to older versions of the firmware (versions prior to V1.1.4). The latest firmware's GUI will let you turn fast-loading on/off via your controller, and supports a much wider variety of game images (.tap, .t64, .d64, .d80, .d81, .d82, .crt, .prg).
I'll still preserve the info below for historical reference.
There has been a lot of discussion on this topic in the following forum post:
I'll try to summarise the cream of it when I get a chance.
The basic gist is as follows:
In order to make use of d81/d82 files in BASIC, you have to utilise a quirk in the c64 mini:
LOAD “$”,8
followed by LIST
LOAD “*”,8,1
followed by RUN
(if needed)