In certain instances, a person who must carry something with them on Shabbos is allowed to do so, but should try to do it by way of shinuy (שינוי), a way different from the norm, in order to minimize the amount and seriousness of prohibitions done*. (see, eg, שמירת שבת כהלכתה לב:נד, about bringing medicine that is critical [but not necessarily urgent])
Since that is so, what shinuyim can be done? I'm looking for practical suggestions on unusual ways to carry an object in public, not necessarily for a halachic discussion; the definition of shinuy is here.
Every person should consult their own rabbis as to whether they are allowed/required to carry with a shinuy**, as well as to whether any of these answers count as a good shinuy, before implementing any of them.
A good answer to this question will be practical, easy, concealable (who wants to look funny?), and not cause pain*** to the carrier.
* For more discussion about the precise definition of "shinuy," see What constitutes a shinui?
** Assuming the situation isn't critical or urgent to saving life, in which case השואל הרי זה שופך דמים (Shulchan Aruch, Orach Chaim 328:2)
*** See my answer