A singularity core (commonly referred to as a D20) is one of the most bizarre objects that can be found in the rings. Appearance wise, it's an icosohedral object roughly 6 meters in diameter, and has two distortion 'tentacles' coming out of it's poles that produce an EMP effect as well as repulse the ship away from them.
Encountering
The singularity can be encountered in one of two ways: floating wild in the rings, or the remains from destroying a hybrid.
Obtaining it from a hybrid is the most common way to find and obtain a singularity, as the hybrid does tend to make it's presence known by attempting to fly towards and zap the player's ship with it's microwave, however given that the singularity inherits all velocity from the hybrid, it is the more dangerous approach to obtaining a singularity, especially when combined with it's other effects.
Finding it in the wild is the safer, but much harder approach to obtaining a singularity. It will appear stationary to the rings, alongside between 2 to 10 initially-stationary iron ore chunks. The singularity itself has no signature, but the iron chunks can make it visible to LIDAR, and especially visible to RADAR.
Behavior
The core exhibits multiple behaviors that tend to it's difficulty to obtain.
Tentacles
The two tentacles coming out of the core's poles are the main focus of the core's weird properties. They visualize where the singularity's weirdness is focused around.
Direct contact with a tentacle exhibits a 5 MW EMP effect on any object, which is powerful enough to knock out any unprotected computer. This effect can be partially mitigated with a physical HUD, or completely negated using a ship equipped with a Faraday Cage.
Repulsion/Attraction
The core exhibits both a repulsive and attractive force on any iron-based object within it's area of effect, which includes both ships and ore chunks alike. These effects are exhibited in radius around any point of the singularity, including it's tentacles, which results in a lozenge-esque area of effect for both forces.
At first, you will notice the attractive effects, which albeit are considerably weaker than the core's repulsive effects, are felt at a much greater radius than the repulsive effects, and drag your ship and any iron ore chunks towards the core. This attractive effect is felt in an area roughly 800 meters long and 400 meters wide, centred and aligned along the core's axis.
The repulsive effects are noticed when close to the core and it's tentacles. It produces a strong repulsive force alongside the EMP effect also exhibited by the tentacles, which you'll likely touch given how close they start to be felt. The effect itself is roughly 4 times stronger than that of the attractive force, and quickly pushes anything away from it. The repulsive force is felt in a significantly smaller area, roughly 180 meters long and 100 meters wide, and also centered along the core's axis.
Capturing the Singularity
Effectively any ship is capable of retrieving a core, given it is able to get it to be considered "captured" at the time that your ship returns to Enceladus Prime, which would involve the core being either attached to a manipulator arm, or considered to be in the cargo bay (being in the excavator's clasp also counts as captured.)
The two problems with capturing the core are the EMP and repulsive effects. The EMP will knock out the computer when not using a ship with EMP resistance or a physical HUD, requiring manual controls to be used, and can be hard even for experienced pilots when faced with the repulsive force. The repulsion is an issue in it's own right, and makes capturing using an arm or within an excavator especially hard. The two ships with a large enough excavator to bring the core into the cargo hold (the OCP-209 and EIME), it can be easier to capture the core as it won't be able to escape once closed around it, however the OCP may face difficulties with it's side-faced door, and the variable geometry assist tuning option may be desired to use the autopilot to chase it.
Once the singularity is captured, you are able to bring it back to Enceladus Prime. If the computer is knocked out via the EMP effect, or rotating the ship for astrogation to Enceladus isn't feasible, there are two other options to bringing it back: Emergency Return Protocol, and exiting the rings. Emergency return involves accelerating the ship past a velocity of 200 m/s, where the ship will bring online it's emergency system and return the ship to Enceladus for excessive speeding. Exiting the rings works in a similar way, except the threshold is either leaving the A-ring, or entering the Encke Gap past 5 km. If successful, you will bring the singularity back with you.
The singularity will sell at enceladus for 1 million Encelado under the title of research material.