I didn't have much luck getting a response to this question before so I have tried to reword and expand it a little:
In early 2010 I attended this inaugural lecture by string theorist- Prof. Mavromatos entitled 'MAGIC strings'. In it he proposes that some string theory models may violate Lorentz symmetry at the Planck scale resulting in a kind-of foamy spacetime that could be observed by differing arrival times of photons of different energies reaching us from distant astronomical sources. See http://www.kcl.ac.uk/news/events_details.php?year=2010&event_id=2178 or here for one of the papers: http://iopscience.iop.org/1742-6596/174/1/012016
Furthermore, in 'Cycle's of Time' that I read recently, Prof. Sir Roger Penrose mentions (page 203) that Wheeler and others have strongly argued that if we could examine spacetime at the Planck scale we would see a turbulent chaotic situation (from vacuum fluctuations of the quantum fields I suppose) or perhaps a discrete granular one. Penrose goes on to list some other approaches that may suggest how this discrete structure may manifest itself. Loosely transcribed these are: spin foams, casual sets, non-commutative geometry, Machian theories, twistor theory, [EDIT] loop quantum gravity, or strings and membranes existing in some higher-dimensional geometry...
I have studied some QM, introductory QFT and the Standard Model as well as some basic GR but I have no formal experience of string theory. My questions are therefore:
What's involved with each of the above approaches? I.e. in what way does the spacetime become discretised? (Particularly in string theory)
Are there any other popular(ish) approaches that should be added to the list?
Supplementary query, with GR being a background-independent theory, I fail to see how one can end up with discretised spacetime without it being a pre-defined background onto which a theory of the dynamics would have to be 'bolted-on'??
Please forgive my ignorance if what I have said is misinformed, all comments and elucidations would be most welcome.