How likely is negative energy to exist? Negative Energy is the exact opposite of ordinary energy as it would move in the opposite direction from its momentum and accelerate in the opposite direction of an applied force.  It would also warp space time in the opposite way from ordinary matter meaning that it would warp space time in such a way that it will produce repulsive Gravity.  If there was an equal amount of positive and negative energy then there would be zero total energy. Negative energy if it exist could potentially be used to stabilize wormholes as well as for warp drive.
How likely is it that negative energy actually exist in our universe?
 A: Here is a flow chart of the forms of energy, with links .

Conservation of energy is one of the fundamental laws governing physical systems and is the only reason why one can talk of "negative energy"
here is a breakdown of the forms that   **Conservation of energy ** takes

In almost all frames negative energy exists, in the sense of conservation of energy, for example between potential forms and kinetic forms. Look at the energy levels of the hydrogen atom, for example, as a consequence of conservation of energy.  It is only in the special and general relativity where the concept of "negative energy" becomes problematic, when it ties up with the mass energy equivalence. Otherwise, negative energy means that, from conservation of energy, positive energy  also exists  in the system, because of the energy conservation law.
Thus the question has  "unexpected"  meaning only if it is asking whether negative masses exist, not in the formulation of your question, where the answer is "yes".
It is only negative masses that would behave the way you describe:

as it would move in the opposite direction from its momentum and accelerate in the opposite direction of an applied force. It would also warp space time in the opposite way from ordinary matter meaning that it would warp space time in such a way that it will produce repulsive Gravity.

All these are results from the existence of negative mass, not negative energy, which exists in all classical systems.
The problem  of whether antiparticles have also negative gravitational mass is under experimental exploration.

We observed the times and positions at which 434 trapped antihydrogen atoms escaped our magnetic trap, and searched for the influence of a gravitational force. Based on our data, we can exclude the possibility that the gravitational mass of antihydrogen is more than 110 times its inertial mass, or that it falls upwards with a gravitational mass more than 65 times its inertial mass.

The experiments have a long way to go.
A: Believe it or not, negative energy actually does exist, in fact, it is all around you. The problem is, it "disappears" almost as soon as it forms. Due to the phenomena of quantum tunnelling, there exists the constant "creation" of virtual particles, and virtual antiparticles. However, these virtual particles and antiparticles "destroy" one another within fractions of a second upon forming. This is actually due to the fact the universe resides in a "False Vacuum State", where the total universal energy constant is not at the lowest potential rest energy. Because e=mc² states that matter can neither be created, nor destroyed, only converted into one form or another, in order for virtual particles to exist, there must form a corresponding virtual antiparticle to carry a negative energy value to ensure a net system energy of zero, otherwise the laws of energy conservation are violated. This is actually how black holes evaporate via Hawking Radiation. As virtual particles forming at the event horizon are ejected from orbit of the black hole, antiparticles are sent downward into the singularity,  taking away energy from the black hole as the virtual particles become real particles. If one were to somehow trap the virtual particle, -such as if the antiparticle escaped the black hole instead- the antiparticle would become real. However, this would hold no practical value, as the now "real" antiparticle would still be neutralized by interaction with ordinary positive energy matter.
