What are the technological challenges to building a muon collider? There is a strong physics case for building and running a muon collider (primarily as a Higgs factory).
Currently, what is the principal technological hold-up in making the muon collider possible?
 A: Making a muon beam is pretty well-established technology. There are currently muon beams at CERN, at Brookhaven, at Fermilab, and probably elsewhere.
It's also possible to inject muons into a storage ring — this is the heart of the muon $g-2$ experiment.
The thing is, though, that the beam in a collider is essentially empty.  At LHC, each proton bunch contains about $10^{11}$ protons.  When two bunches overlap in a detector, there are a few dozen proton-proton collisions.  But the remaining protons are free to go around the ring and collide again.  I think that most of the storage-ring colliders fill the ring with protons (or heavy ions) at the start of a data-taking period and then re-use those protons for days or weeks.
You wouldn't get to re-use the muons very efficiently.  A muon at rest lives for 2 μs; even boosted to 14 TeV ($\gamma = E/m \approx 130\,000$) the lifetime is less than a second.  So you'd have to be continuously producing new muons and continuously accelerating them prior to injection — much more like a rare isotope facility than a high-energy collider.
There would also be a continuously-produced beam of electrons from the muons decaying in flight.  For slow particles, the bending magnets would separate the muons from the electrons.  Boosted to a few TeV, however, the muon and its decay electron have essentially the same momentum, and your muon beam would be replaced in the storage ring by an electron beam with slightly larger momentum spread.  This beam would have to be dumped somewhere — again, probably about once per second.
I think it's probably doable, but it'd be quite challenging.  Give it ten or twenty years.
A: The biggest technical challenge preventing a muon collider is how to safely generate and transport the beam from the source into the storage ring. Generating muons, unlike other particles like electrons or protons, makes a really messy beam with a very large transverse momentum spread. The "emittance" of a beam is roughly defined as $\varepsilon = \sigma_x^2 \sigma_{x'}^2$ to within some normalization conventions. A storage ring has a certain "acceptance" which is the largest emittance beam it can store without particles hitting the beam pipe or otherwise getting scraped out of the beam.
The trouble is that you generate a shower of muons by hitting a fixed target with high energy protons. The resulting muons then have to be gathered and cooled to a small enough emittance that you can get them into the storage ring without violating radiation safety restrictions on how much beam you can lose. When those particles hit the beam pipe, they generate a shower of neutron radiation, so you have to keep that down. In fact, the initial design of a muon collider would have led to so much radiation that the phrase "Ring of Death" was used to describe the problem. Thus, the beams have to be cooled, so their emittance is small enough to get into a realistic storage ring.
Normally what is done, such as with anti-electrons which are produced in basically the same way, is the particles are put in a very large acceptance damping ring where you sit and wait for the synchrotron radiation to cool off the bunch transversely, or use other methods such as stochastic cooling or electron cooling. Muons have a second problem, which is that they have a half-life and the beam itself will decay.
So you have to take a very messy beam of particles and cool them off before you lose the entire beam to its decay into an electron and some neutrinos. Nobody has figured out how to cool them fast enough while still holding onto enough of the bunch to make the experiment worth doing.
