Q:

What are some approaches to discrete space-time used in modern physics?

This thought gave rise to some new questions in my mind.
What are the consequences for:

How would it affect duality i.e. particle, wave property of photons?
How does this statement affect the information theoretical aspect (entropy) of the universe?  Update: Given a volume V of space, is the entropy (maximum information that can be store) in this volume changed when this statement is applied?
How is a black hole affect by this statement? Update How is entropy changed inside the black hole?
Could one consequence be that the universe is hologram, since the construction isn’t continues?
Would the smallest quantified space be planck's constant? Is there an equivalent constant for time?

I hope to get some of your feedbacks regarding this statement.

A:

Let's try and make things more precise, step-by-step.

There's no such thing as "particle-wave duality": the name-of-the-game is "Quantum Field Theory". This paradoxical notion of a possible "duality" only happens when you don't use the appropriate framework to describe your Physics. Therefore, it makes no sense to speculate on what would happen if spacetime were quantized/discrete: in this scenario, the question would be: "Would a quantized/discrete spacetime affect Quantum Field Theory?" And the answer to this question is "No." The reason being that different physical theories have different domains of validity, given by the characteristic energy of the phenomena they describe.
What is the "information theoretical aspect of the universe"?! This is not even appropriately defined, let alone "well defined".
The black hole is the stereotypical object in a quantum gravity theory. So, when you quantize spacetime, you should look at black holes to see what happens. We already know that black holes have Entropy. So, the very first question should be: What does your particular quantization scheme yields for black hole Entropy? The current state-of-the-art, as far as i know, is that all different schemes of quantization of spacetime yield a reasonable answer to this question.
This question, again, is not even appropriately defined, let alone "well defined". Holography has a very precise and well defined meaning in Physics, which is not related to the hologram in a credit card, for example. So, holography does play a role in quantum gravity, the more famous statement being that of AdS/CFT. But, as it stands, your question does not have meaning.
This has already been stablished a long time ago: if you quantize spacetime, the smallest unit of spacetime is given in terms of Natural units.

