There are 3 main experiment types ongoing in the search for a particle candidate of dark matter.
First is what is known as "direct detection", whereby you infer the existence of dark matter in our galaxy by it's interactions with a large body of material (your detector). Typically these are noble liquid and the dark matter interacts with the nucleus of these liquid. The LZ experiment is an example of a direct detection experiment,
Second is known as "indirect detection", where you try and observe the photons produced from the self-annihilation of dark matter in our universe (similar to electron+positron annihilations). FermiLAT is an example of such an experiment.
Lastly is "collider experiments", where you collide particles in a particle accelerator with enough energy to produce dark matter. The detectors aren't able to detect the produced dark matter, instead they infer the existence of dark matter from the imbalance in momentum (should be balanced due to the conservation of energy). The Large Hadron Collider (LHC) has a few results for such searches.
I am not so familiar with experiments on dark energy.
I'm part of a research group that is using a crowd-sourced supercomputer to constrain the distribution of dark matter in the Milky Way.
Among other things, part of the project is to run tons of simulations of tidal disruption of dwarf galaxies by the Milky Way (similar physics to why we get tides on Earth) and then match them to what we see in the night sky in order to see how accurate the simulations are. The simulations have varying dark matter models. From there, you can match parameters of the simulations to what those values must be in real life, which will give us some information on how the dark matter in the Milky Way is distributed. There's a lot more to it and since my research isn't on the dark matter side of things I'm not as knowledgeable on the topic as other people in the group. I know some of them are redditors and might stumble on this to give better information.
If we know how the dark matter is placed in the Galaxy, then we can know how it collapses/dissipates and are one step closer to finding out what actually makes up dark matter and its properties.
Beyond this, there are plenty of experiments that look for interactions between dark matter and large underground tanks of noble liquids, some of these were explained in this thread already. The only experiments I've heard of on dark energy are looking at the distribution of mass in the universe and determining if large scale structure is caused by differences in dark energy in locations (which would speed up/slow down expansion in those areas), tiny energy differences in the cosmic background radiation and density differences in the early universe or some combination of all this.
One experiment I am familiar with in the search of dark matter is the [PICO collaboration](https://en.wikipedia.org/wiki/PICO
They utilize bubble chambers underground at SNOLAB in Canada. When energy is deposited from an interaction, a gas bubble is formed that can be detected visually by cameras and acoustically via piezoelectric sensors (basically very sensitive microphones). If they can eliminate all known background sources, then they may be able to conclude that a recorded interaction was from dark matter.
An interesting side note, SNOLAB was the location of the first experiment to directly demonstrate oscillations of solar neutrinos, for which the 2015 Nobel Prize for Physics was awarded. If PICO is successful, it could lead to the second Nobel Prize for a direct detection experiment at SNOLAB.
When it comes to studying dark matter. You have three options: shake it, break it, or make it.
Shake It, otherwise known as "direct detection", is whereby using some well isolated, dense, and very sensitive detector, you hope that dark matter will interact with it. As the solar system moves through the halo of dark matter that bathes the galaxy and the earth around the solar system, we expect that the flux of dark matter through our detectors will go up and down with an annual period. This is the kind of signal we expect IF we were to detect a signal in the detectors. such detectors include XENON1T, LUX, LZ, ADMX, PICO, and many others. These experiments typically use very dense materials which give off some light or charge when their atoms experience some recoil, or "get shaken". ADMX is the exception since it uses magnetic fields to try to detect axions. (but that's a topological story for another day)
Make It, otherwise known as "collider experiments", is whereby colliding high energy particles in an accelerator, you hope that dark matter can be created in the decay of some intermediate particle. Since dark matter wouldn't interact with the calorimetric detectors, you analyze the "visible" collision byproducts and look for missing energy that can't be accounted for by neutrinos. Such searches take place at accelerator facilities like the LHC at CERN and (before 2011) at the Tevatron in FermiLab.
Break it, aka "indirect detection", is whereby using astronomical particle detectors, you hope to detect the decay or annihilation products of dark matter. Since we can observe concentrations of dark matter by studying things like gravitational lensing, we expect that these annihilation and decay signals would be strongest in those direction. For example, we know that dark matter is densest in the vicinity of galactic cores, dwarf galaxies, and the center of galactic clusters. Hence we point gamma ray, neutrino, and cosmic ray detectors in those directions hoping to pick up those signals in excess of what we might otherwise expect. Such excesses have been reported coming from our own galactic center in gamma-rays. Detectors that are used for these searches include Fermi-LAT, VERITAS, MAGIC, HESS, HAWC, AMS, IceCube, ...etc. In this category, any detector that can detect particles coming from space with some kind of directional discrimination can be used. There are some common themes though. For example, certain popular models (WIMPs) expect dark matter to be above several MeV in mass, thus we expect that any electromagnetic annihilation signature would be in the gamma-ray spectrum. Thus we use Fermi-LAT, VERITAS, MAGIC, HESS, HAWC, which observe gamma rays from several MeV to several hundred TeV! Likewise, we expect if neutrinos are generated, to be of very high energies so IceCube is used to look for those.
Experiments: XENON1T, LUX, LZ, ADMX, PICO, LCH, Fermi-LAT, VERITAS, MAGIC, HESS, HAWC, AMS, IceCube
) showing the three ways to detect dark matter (X) with normal matter (q)