At what point does a singularity become so massive that not even gamma rays can escape its influence

First of all a singularity is a point where all laws of physics and maths break down. Secondly a mass (any mass) needs a certain radius (called the Scwartzchild radius) where light can't escape. This radius is given by the very simple formula: r = 2GM / c^2

Here's the derivation of the formula:

Escape velocity is the velocity needed to escape the gravitational influence of an object. It is given by the formula Ve = sqrt(2GM/r). Squaring both sides. Ve^2 = 2GM/r, multiplying by r and dividing by escape velocity r = 2GM/Ve^2 Now if we want the radius where the escape velocity = the speed of light we just replace Ve with c so we get r = 2GM/c^2. It's just that simple.

I think you're a little confused because you can detect gamma rays from black holes so you think somehow the gamma rays are escaping the event horizon even though the escape velocity is greater than the speed of light. This isn't the case. Gamma rays travel at the speed of light just like radio waves or visible light. The gamma rays that are detected are from the acretion disk around a black hole, think of it as the dying scream as matter is ripped apart by gravitational sheering as it is spaghettified just prior to plunging into the black hole.

Hope this helps.

Nutrinos and dark matter have mass and therefore travel slower than the speed of light therefore a radius even slightly larger than the Scwartzchild radius would keep nutrinos and dark matter gravitationally bound. All youneed to do is take their velocity and plug in the equation Ve = sqrt( 2GM/r). Nobody knows what dark energy is, so your guess is as good as mine. Tachions are hypothetical particles that travel beyond the speed of light, I don't know how fast they go but the limit to what the radius of a black hole would have to be to trap them is given by the formula r = 2GM/Ve^2.

Once you know the velocity of something plug it into that forumula and you'll know the radius a certain mass has to be crushed into in order to have high enough gravity to trap it.

You are a bit confused. First of all, the wavelength of the electro-magnetic radiation doesn't affect the behaviour in relation to black holes. Radio and gamma photons are both trapped in the same way.

Second, the small dense mass becomes a black hole at the point where it becomes a singularity. So any singularity traps photons.

All electro magnetic radiation obeys the light speed limit and behaves the same way in the presence of a singularity.
All stop (or are infinitely red shifted) at the same moment and the same time.

The point at which the mass of an object is contained completely within its Schwarzschild radius, 1-2M/r = 0.