My guess at a list of them would be: spin foams, casual sets, non-commutative geometry, Machian theories, twistor theory or strings and membranes existing in some higher-dimensional geometry...
Firstly, what have I missed?
Also, if there is a string theorist or similar on here that could briefly shed some light on what's involved with these approaches I'd be most grateful.
[Editted/expanded from here on]
OK so I'm not having much luck getting a response to this so I suppose I've not been clear enough. I'll try to expand a little:
In early 2010 I attended this inagural 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, as I mentioned in the comment below, Prof. Sir Roger Penrose mentions that Wheeler and others have strongely argued that if we could examine spacetime at the Planck scale we would see a tubulent chaotic situation (from quantum-mechanical vacuum fluctuation I suppose) or perhaps a discrete granular one. Penrose goes on to list some other approaches that may suggest how this discete stucture may manifest itself. Hence my list above.
I have studied much QM, introductory QFT and the Standard Model as well as some basic GR but I have no formal experience of string theory. I therefore wondered if anyone here could jot down a sentace or two to explain what's involved with each of the above approaches and add to the list if I've overlooked some alternative popular(ish) approaches.