Technically, we don't find gases in water. "Gas" is a state of matter, so a little pocket of gas could be surrounded by water, but then we call it a bubble. Instead, what we find is that some substances like oxygen or carbon dioxide, which would be gases under normal conditions, can be dissolved in water. But then they're not gases anymore, they're part of a liquid mixture or solution.

Plasmas would be the same as gases. Plasma is a state of matter which is so energetic that some or all of the atoms have had their electrons stripped off. Plasma is made of free electrons and the positively-charged atoms (ions) which are left behind when an electron leaves. This is too energetic a state to coexist with water. Either Plasma would boil the water, or the water would cool the plasma, or else the plasma would occur only within a bubble.

People who do underwater welding sometimes can sometimes generate Plasma under some conditions. But that only means that the water immediately surrounding the torch is boiling away faster than the water can cool the plasma. So there's actually no plasma in the liquid water itself, although the steam (gaseous water) can become part of the plasma. Otherwise the plasma just generates a bubble.

I've looked around a bit and can't find too much information for or against.My best guess would be that it is not possible, since plasmas have much more energy than gases, and it would probably be very difficult to keep them stuck in the water. However, it is possible to have solid particles stuck inside a plasma, in industrial processing plasmas or space plasmas. These types of plasmas are called dusty plasmas. This would be another example of a colloid, which is when one state of matter is suspended in another state of matter - just like having a gas in water (a liquid).

Plasmas are by far the most common phase of matter in the universe, both by mass and by volume. All the stars are made of plasma, and even the space between the stars is filled with a plasma (a very sparse one). Common forms of plasma exist within gas phases such as the polar aurora and lightening. There are also lots of astrophysical plasmas like stars etc. Industrial plasmas include plasma displays (like Plasma TVs) and inside fluorescent and neon lamps. However, I dont know of and was not able to find any examples of plasma within a water phase.

No.

First off, gasses dissolve in water, but when they are dissolved they aren't technically gasses. They are dissolved solutes that become gasses if released.

Plasma is ionized matter, and to ionize matter you need temperatures of at least 3000K (about 5000Fahrenheit!). The boiling point of water is 373.15 K.Now, the boiling point does increase if you put water under pressure, but the water is still going to be lower temperature than the plasma. If the plasma is hot enough, it will ionize the water around it and make more plasma. This of course will cool the plasma off, and pretty soon it's not going to be hot enough to be ionized anymore, and it will stop being plasma.