There aren't exactly an abundance of stealth planes, but most highly developed countries own one. Why are there no stealth helicopters? If there are, why are they so obscure?
Well there actually are stealth helicopters, from Boeing Sikorsky’s RAH-66 Comanche, to a classified low observable derivative of the UH-60 Blackhawk, which was used in operation Geronimo to kill Osama bin Laden. They’re out there but haven’t seen the light of day for several reasons.
UPDATE: The US Army has procured a new scout / light attack helicopter program with the Bell 360 Invictus program under development. The Invictus, a two seat attack helicopter which incorporates low observable features appears to follow many of the configuration choices made with the RAH-66 nearly two decades ago.
In the case of the RAH-66, It was development problems and massive cost overruns which led the secretary of the Army to cancel that program. It would have been interesting to see what the production version of the Comanche had in terms of an RCS, as with the electronics it carried plus the integration of the brimstone anti-tank missile would have made it very formidable.
It’s possible as well that with the Comanche and the classified Blackhawks that the military has had a chance to evaluate these systems and determine that low observable applications and helicopters was either not beneficial for their domains of operation or simply superfluous. In the case of the classified Blackhawks, those items are sort of delta force type weapon system used for special, clandestine missions and there may not be a large need for a fleet of these things.
Given most military helicopters are operating nap of the earth, and the major threat to them comes in the form of small arms fire at visual distances, low observable technology (at least for radar) really isn’t much help in this situation. Helicopters do make use of some low observable technology, particularly IR suppression systems for the engine exhaust, to make them less vulnerable to MANPADS, but radar detection really isn’t a big threat to them, as they generally operate well under the observable ranges of AAA and SAMs used in IADS. A Zeus, however, is a very dangerous thing to face down in a helo; it’s possible that LOT may be of help here.
The closest question to this currently is Why aren't modern helicopters quieter?, which one should read first. That regards sonic stealth.
As for radar stealth, any rotating part, visible from the outside, is a major source of unavoidable radar reflections. In 4th generation jet fighters, the engine faces generally contribute more than half of the total RCS (radar cross-sections). Very simple tweaks like installing radar blockers into the intakes can reduce the RCS several-fold.
The rotor is a very large rotating part. Even if made completely radar-transparent, the rotor will accumulate dirt and grime from the air, which have some reflectivity. On a properly stealth-shaped surface, this will be deflected away from the mainlobe. But a rotor, constantly rotating and flexing, has an uncontrollable shape.
This means helicopters can never reach B-2 or F-22 level of stealth. They can get lesser RCS reduction, along the lines of the Silent Eagle proposal. This could be useful, but helicopters fight at much shorter ranges than fighters or bombers.
The F-15 is expected to be visible to high-end opposition fighters out to 300-350 km, and the F-22's stealth reduces that to about 40-50 km against most fighters. At that range, radar stealth is defeated by more powerful or longer-wavelength radars, and even an aircraft with no radar cross-section could be detected by IRST/FLIR systems (correspondingly, high-end opposition fighters use all of these). The F-22's weapons have some hit probability at ranges up to ~80 km against fighters and up to twice that against other aircraft, creating useful room to shoot first. Helicopter radars top out around 50 km and their weapons even closer.
The Comanche could never hope to be as stealthy as the F-22. It's hard to guess its best-case scenario, but something like 60-100 km would've been pretty good. Operationally, for a helicopter, such detection range doesn't make much difference.
Helicopters can already hide from THAAD (theater area air defenses, mostly networked long-range SAM such as the Patriot or S-400) by nap-of-the-earth flying, placing surface features between them and the radar, and hiding among the ground clutter. It's only at 15-40 km that a helicopter would come in direct view of short-range air defenses (highly mobile "all in one" transporter, launcher and radar vehicles). A Comanche would still be visible and targetable by radar at this range.
Essentially, a helicopter's stealth would help protect it from enemy fighters. But there's more than one way to protect your aircraft from enemy fighters.
For the US, it's air superiority. The US maintains an air force that, including the USN, is a match for the rest of the world combined for air-to-air combat. It's not by happenstance - US military doctrine prioritizes air superiority, so resources were spent on it.
With air superiority, a helicopter doesn't need to hide from enemy fighters; the risk they face in every mission won't be worth it for such a low-priority target. The Comanche project was started in a wave of excitement with stealth, with an assumption that stealth could be an easy add-on to it. Without stealth considered, it was underwhelming. Radar-stealthing a helicopter proved difficult and expensive, and there just wasn't much to be gained by it.
Well, there is the RAH-66. Stealth helicopters are difficult to make because of the radar cross-section of the blades. Even on the B-2 the fan blades are shielded so they don't create a massive return, and helicopter blades would create a much bigger return. Even with the Comanche's FENESTRON tail rotor, it's not very quiet. Helicopters are loud machines, no matter how you slice it. There may be quiet helicopters, but they're still pretty loud. To answer your question about obscurity, well, it''s like the B-2 or SR-71. Nobody wanted anyone else to know it existed because the tech was so valuable. Same with stealth helicopters.