The Bug:

You can create illegal flows of water quite easily using any obstruction not destructible by water, for this report signs have been used, and two adjacent blocks that are destructible by water, this would include air.

Bug has existed since official release, found instance of bug in Alpha v1.0.4 and Alpha v1.0.5_01, and bug continues to persist in all modern versions. As this bug has been in the game for such a long time, though it does follow a similar acceptable flow to another case shown later in report, it is most definitely not following the same logic.

Steps to Reproduce:

To begin said glitch, place water and sign to get water flowing like so. The block placement is also important due to how water flows in game. Behind the signs, it is important to note that water recognizes that air is a destructible block by water, and hence the glitch will work. This was found through methods such as placing other blocks that are destructible by water and the glitch working. For demonstration purposes to keep the area clean, air will be kept as the destructible block.

Next, you will want to place either two signs to obstruct the water or one block and a sign to obstruct the water. Place two signs on the front of the signs that were next to the water flow.

Following the last step, you will want to place two block away from the structure and then place a sign on the top block.

This is how the structure should look before the glitch is finally performed.

You may now break the two signs obstructing the water, starting with the top sign.

This causes the water to now not flow properly, instead acting like a similar case of an acceptable flow of water which will be shown later.

The rest of the blocks, aside from the two next to the water flow, can be broken and the illegal flow will continue to exist.

 

Now, these illegal flows can flow out in all directions from a source as well.

4 Flow 7x7

These four directional flows can do the glitch all the way up to the water flow of seven, though the seventh water flow changes slightly, but the glitch works nonetheless.

5 Flow 9x9

6 Flow 11x11

7 Flow 13x13

You can also have the illegal flow go in just two, or three directions, instead of just one or all four.

Similar Acceptable Case added in 1.13:

Now for the similar acceptable flow of water that this glitch applies to, it seems to follow the logic of waterlogged blocks, but as stated earlier, this glitch has been in the game since early Alpha, which means that cannot be the case.

As shown above, when water goes to flow over a water logged block, it doesn't continue to flow, but instead stops in its tracks. This is very similar to the glitch happening and I only put it here as a reference point of a similar acceptable flow of water.

 

Video Link to YouTube Video of Bug being performed:

https://youtu.be/v2FJMgCvCo8