The Gossamer Albatross is a human-powered plane with a top speed of 29 km/h (18mph).
It was used to cross the English Channel and seems to meet the criteria of the question.
aviation.stackexchange.com/users/22749/Michael Schumacher
The Antonov AN-2 has no stall speed quoted in the operating manual and can fly under full control at about 30 mph.
Thus if the headwind is sufficiently large the aircraft will move backwards with respect to the ground.
aviation.stackexchange.com/users/15202/Brilsmurfffje
The Harrier , Yak-38 , Yak-141 , XV-15 , and V-22 are all fixed wing aircraft.
All can hover in mid air, controlled.
So they are in controlled flight at 0 velocity.
At least the Harrier can even be in controlled flight flying backwards, so with negative velocity.
The others may as well, I don't know.
aviation.stackexchange.com/users/900/jwenting
now, if you're looking at modern, more commonly used transportation, powered paragliding would probably take the cake.
CC BY 3.0, https://en.wikipedia.org/w/index.php?curid=13110495
Powered paragliders usually fly between 15 and 50 mph (25 and 72 km/h) at altitudes from 'foot-dragging on the water' up to 24,000+ ft (5400 m) https://en.wikipedia.org/wiki/Powered_paragliding
The beginners equipment are often the slowest, and with proper skill, and the right weather, can have a stall speed of zero .
Whilst being lifted by a thermal, the only thing that will push the glider forward is the natural tilt.
naturally, on a still day, the slowest of equipment will stall at below 10mph .
Disclaimer: this is not an airplane, but it is a fixed wing vehicle, made to primarily move forward to generate lift, akin to most airplanes.
aviation.stackexchange.com/users/11387/tuskiomi
If you are including historical aircraft, the Wright Flyer averaged 10 fps (approx 6.8 mph or 11 kph) over
it's first 120 foot flight in 1903.
Reference: https://airandspace.si.edu/exhibitions/wright-brothers/online/fly/1903/triumph.cfm
aviation.stackexchange.com/users/22751/user112358
Building it just for very slow speed does not look practical (if you really need this, use helicopter).
Some old planes may be slower, but they do not use the newest technologies and may not be built for slow flight anyway.
Some patrol, agricultural planes may benefit from slow flying, but we need something more extreme.
I expect such aircraft to be some specific machine that has a huge wing and low weight for other reasons, and should be recently built to benefit from latest technologies.
Solar impulse looks like a good candidate.
It has take-off speed of 35 km/h (22 mph) only.
Looks like
its minimal speed is about 20.67 mph only.
Its large wing holds the solar batteries.
It has a wingspan larger than Boeing 747 and the weight close to Cessna 172 !
(picture from Wikimedia Commons ).
aviation.stackexchange.com/users/917/h22
The Ruppert Archaeoptrix Electro ( Wikipedia , official website ) apparently has a stall speed of 30 km/h (19 mph / 16 kn) , and I think that makes it a candidate for the current "slowest" fixed wing aircraft.
It is a relatively new glider that can be foot launched, but there are also wheeled and motorless configurations, and it can also be launched by towing. For the motorized version:
aviation.stackexchange.com/users/17739/flawr
Gossamer Condor , the first human powered airplane capable of basic maneuvered flight.
When it finally won the Kremer prize for a 1 mile figure 8 course
it did that course in 7 minutes and 22 seconds.
The later Gossamer Albatross that crossed the English Channel had to be faster.
This assumes that the question criteria was AIRSPEED (not groundspeed) for a fixed-wing, man-carrying aircraft that could take off and land under its own power and
maneuver both into and out of the wind (figure 8 maneuver).
aviation.stackexchange.com/users/22756/pbm
You should check out planes with custer wings.
https://en.wikipedia.org/wiki/Custer_CCW-5
It was claimed that the aircraft could fly under control at 11 mph (18 km/h) and that it could take off with a 1,500 lb (680 kg) load at 70% power in 90 ft (28 m).
aviation.stackexchange.com/users/22771/Foo
Slepcev Storch
(A 3/4-size replica of the Fieseler
Fi
156 Storch , a German WW2 reconnaissance aircraft well known for its slow-speed performance) 36 km/h or 40 km/h (19kn/22kn) Picture Source Performance according to Wikipedia :
Maximum speed: 150 km/h (93 mph; 81 kn)
Cruise speed: 133 km/h (83 mph; 72 kn) Stall speed: 40 km/h (25 mph; 22 kn) Rate of climb: 6 m/s (1,200 ft/min) Performance according to ulm.it : Maximum speed: 150 km/h VNE: 182 km/h Stall speed: (flaps out) 36 Km/h Climb rate: 9,1 m/s (1800 ft/min)
aviation.stackexchange.com/users/790/Thunderstrike
I was at the Biggen Hill airshow many years ago where not one but two different aircraft flew backwards.
One was the AN-2 which was able to nose up into the headwind so far that it actually flew backwards for a short distance down the runway.
The other was a Russian super jet, (possibly a Sukhoi 27 or Sukhoi 35 ?)
that did a vertical climb on afterburners that shock the ground then slowed to a stop before falling on its tail while still vertical.
It flew tail first towards the ground before lowering the nose.
aviation.stackexchange.com/users/2233/Paul Smith
Alaskan Bush Planes are modified to fly very slowly for short landings: Features include:"Short runway requirements, typically gained through high aspect ratio wings and high-lift devices such as flaps, slots and slats to improve low speed flight characteristics, allowing shorter ground rolls on landing or takeoff."
https://en.wikipedia.org/wiki/Bush_plane
This might be helpful too: https://en.wikipedia.org/wiki/List_of_STOL_aircraft
aviation.stackexchange.com/users/22813/Gues
Here is a video showing that the Gossamer Albatross flew at lowest speed, 7mph (11.2kmph).
Very slow.
But the save speed will be 10mph (16.1kmph), as mentioned in that video.
It lasted 2 hours to cross England strait from England to France.
aviation.stackexchange.com/users/34998/AirCraft Lover
It's easy to find information about the fastest airplanes, in different categories (e.g.  X-15 ,  SR-71 , the  Concorde  etc), but what is the slowest one? Which powered, manned airplane is capable of sustained level flight at lowest velocity?
What is the slowest fixed-wing airplane?
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<aviation-history><airspeed><fixed-wing>
aviation.stackexchange.com/questions/39327
aviation.stackexchange.com/users/8195/el.pescado
There are a number of candidates for the slowest fixed-wing airplane. Candidates include the Antonov AN-2, the Solar impulse, Alaskan Bush planes and the Gossamer Condor. Without there being a clear winner, the candidates that have been built with these specifications in mind are: the Gossamer Albatross, the  Antonov AN-2, Solar impulse, the Ruppert Archaeoptrix Electro, the Gossamer Condor, the  Slepcev Storch and Alaskan Bush Planes.
Without there being a clear winner, the candidates that have been built with these specifications in mind are: the Gossamer Albatross, the  Antonov AN-2, Solar impulse, the Ruppert Archaeoptrix Electro, the Gossamer Condor, the  Slepcev Storch and Alaskan Bush Planes.