Of spacecraft missions to minor planets, three are active:
New Horizons now at about 50 AU from the Sun.
Hayabusa2 successfully delivered the samples that it collected from the asteroid 162173 Ryugu. It went into orbit around the asteroid in mid-2018 and surveyed it for the next year and a half, deploying landers and rovers and collecting samples. It then went into an orbit with a flyby of the Earth and it released a capsule that contained its samples. That capsule successfully landed, and the spacecraft stayed in orbit around the sun. It will fly by some asteroids in coming years.
OSIRIS-REx is returning the samples that it collected from the asteroid 101955 Bennu. It went into orbit at the end of 2018, surveyed the asteroid, landed on it late in 2020, collected up some samples of it, returned to orbit, and departed for Earth on May of this year. Its sample capsule should arrive at the Earth in September 2023, and the spacecraft should continue to another asteroid, currently 99942 Apophis, one with an orbit that gets it dangerously close to the Earth.
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Double Asteroid Redirection Test - DART for short - is scheduled to launch on 24 November 2021. It will impact asteroid 65803 Didymos a year later, as a test of a planetary-defense concept.
Psyche (spacecraft) and
Janus (spacecraft) will be launched together in August 2022.
Psyche will fly by Mars nearly a year later for a gravity assist, and it will go into orbit around asteroid 16 Psyche early in 2026. This asteroid is largely metallic, likely the largest asteroid with that composition.
Janus is two spacecraft, one for (175706) 1996 FG3 and one for (35107) 1991 VH, both binary asteroids. They will both do a gravity-assist flyby of the Earth in 2023, and fly by their targets in 2026.
Near-Earth Asteroid Scout is one of 13 cubesats that will be launched atop the first Space Launch System rocket to launch, likely next year. It will deploy a solar sail, then do gravity-assist flybys of the Moon to escape into heliocentric orbit. Once there, it will fly by some asteroid, currently 1991 VG.
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There are several missions in various stages of planning and proposal, but I won't go into those.