I'm no sports injury specialist however, I do take an interest in what may or may not benefit me when i'm broken. I'm currently pondering where to go with giving my wonky ankle some support during it's rehabilitation. Enough support that I can be active on it too. So I have been looking at taping with a particular focus as to go with Elastic or None Elastic tape to aid my recover. 

 Elastic Tape v None Elastic Tape

                           V

The main focus for information has been from a research report prepared by: 

KRISTIN BRIEM, PT, PhD1 • HREFNA EYTHÖRSDÖTTIR, PT2 • RAGNHEIDUR G. MAGNÚSDÓTTIR, PT3 RÚNAR PÁLMARSSON, PT3 • TINNA RÚNARSDÖTTIR, PT3 • THORARINN SVEINSSON, PhD4 titled: Kinesio taping in treatment and prevention of sports injuries: a meta-analysis of the evidence for its effectiveness

Some of the information contained within I found insightful. The Study (mentioned above) used Kinesio Tape (KT) tape, a None elastic sports tape and no tape on lower limbs and compared. 

Kinesio tape is purported to (essentially):

•  Improve local circulation
•  Reduce edema
•  Facilitate or relax muscle
•  Improve joint function by enhancing sensory mechanisms.
• When used to prevent ankle sprains, it may be better tolerated (as it's latex free) and more cost effective (as it can be left in place for 3-5 days) than taping with nonelastic athletic tape.

Here's the crunch: In terms of aiding and stablising an injured ankle: Due to its elastic properties, the ability of KT to enhance functional stability of the ankle relies on its purported effects on proprioception

and muscle activation rather than mechanical support. (Importantly however up to May 2011 this had not as yet been tested). The results conclude: Ankle taping with nonelastic athletic tape plays a role in providing support for the ankle and may further enhance muscle response by maintaining greater levels of muscle activation. The hypotheses as to the effects of KT were not supported by the data. 

KT has further been proposed to provide positional stimulus through the skin and sensory stimulation. The results indicated however that no facilitation occurred and that KT neither affected muscle activation nor resulted in an improved sense of stability for athletes with ankle instability. Therefore, the role of elastic tape, such as KT, with respect to further prevention of inversion ankle sprains, is unsubstantiated.

KT did not alter muscle activity in healthy males before, during, or after a sudden inversion perturbation, while balancing on a tilt board. This is in contrast to the nonelastic tape condition, in which significantly greater muscle activity was seen, as compared to an un-taped condition. The role of KT with respect to prevention of ankle injuries still needs to be substantiated. I'm not sure if it has to date. 

My observations then is that for stability and support on an acute injury then none elastic tape appears to offer more benefit in particularly restricting some of the movement and further by encouraging more/increased muscle activity. Elastic tape appears to offer different benefits (outlined above) however the claims are not as yet substantiated. Is there merit in combining the two i.e to stablise with none elastic tape (as an anchor) and then to use elastic tape which may assist in reducing edema and improve circulation around the injury?

References: 

Kinesio taping in treatment and prevention of sports injuries: a meta-analysis of the evidence for its effectiveness.

Effects of Kinesio Tape Compared With Nonelastic Sports Tape and the Untaped Ankle During a Sudden

Inversion Perturbation in Male Athletes: http://www.jospt.org/members/getfile.asp?id=5228

Does Kinesio Tape Work for Runners? A Hard Look at the Science and Research

CRAZY TAPE ON OLYMPIANS: DOES IT WORK?