At the moment I can only set this to 'all file types' or 'images only' but would like to specify a small list (.doc, .docx, .pdf) for some documents, is this possible?
There is no native way to do what you want. It's either "images" or "all".
If one does not already exist then creating a custom fieldtype that would only permit uploads of a particular type should be possible. Start with the existing file fieldtype and see if you can modify it to do what you need.
you can determinate the allowed file type in EE from this file ./system/expressionengine/config/mimes.php
and be sure that Admin -> Security and Session Preferences, "Apply XSS Filtering to uploaded files?" set to "yes".
and this will apply to all member groups except Superadmin, superadmin members always can upload all files type.