At the current time, is SHA256 the de facto standard for strong cryptographic hashes? At the current time, is SHA256 the de facto standard for strong cryptographic hashes?

From what I am seeing of more sites utilizing it, I would suppose the answer to this is yes, but would like to hear what the community here has to say about this and why it is happening.
Yes, SHA-256 is currently the de facto standard. The "why" is that MD5 and SHA-1 are unsafe and the only algorithm left which has been extensively studied and deployed and which nobody has found a significant attack against is SHA-256. (There is also SHA-512, but people seem to regard it as being overkill.)

There is widespread uncertainty about whether someone will discover an efficient attack on SHA-256 in the near future. It was this fear which motivated the SHA-3 competition.

You should also consider using a generic length-extension defense such as the "SHA-256d" design by Ferguson and Schneier. SHA256d(x) = SHA256(SHA256(x)). If you don't use a generic length-extension defense then you have to consider the consequences of length-extension attacks in each place that you use a hash function. Such considerations can be tricky, and if you misunderstand one, this can sometimes cause a security vulnerability. This is why it is safer to use a generic length-extension defense every time you use a hash function.

If for some reason you don't want to use SHA-256, and you don't want to use a new hash function such as a SHA-3 candidate, then you might also consider Tiger. It has the distinction of being one of the longest-serving hash functions for which there is no known practical attack. However, it has been seriously wounded (i.e. by impractical but theoretically significant attacks), and I would consider SHA-256 safer than Tiger at this point. If efficiency matters, Tiger is faster than SHA-256, SHA-1, or any SHA-3 finalist on 64-bit hardware. It doesn't come with generic length-extension defense built in, but you can of course define Tigerd(x) = Tiger(Tiger(x)).

I would also regard the current crop of SHA-3 finalists as very likely to be secure: http://ehash.iaik.tugraz.at/wiki/The_SHA-3_Zoo . They are much newer, of course, than SHA-256, but they were all designed with a thorough understanding of the flaws in MD5 and in SHA-1, and they have been subjected to intensive scrutiny for almost three years now. They all come with generic length-extension defenses. My favorites are Blake and Skein, which are faster than SHA-256 on Intel/AMD chips. Unfortunately all of the SHA-3 candidates are slower than SHA-256 on ARM chips.

Don't use MD5 or SHA-1. Please.