Only the last part of this question is amenable to a straightforward answer. All wavelengths are potentially observable using different techniques, from gamma rays through to very long wavelength radio waves, over at least 13 orders of magnitude in $\lambda$.
Some of these wavelengths - Gamma rays, X-rays, UV, far infrared - require space-based observatories because the respective wavelengths do not penetrate the Earth's atmosphere - e.g. Integral, Chandra, IUE, Herschel. The rest can take place on Earth, though there are advantages in doing these from space too, either because the atmosphere blurs the image (optical - HST), or because Earth-bound interference can be a problem (radio wavelengths).
The biggest factor that affects how much detail can be resolved is the aperture of the instrument. The best angular resolution in principle is $\sim \lambda/D$. This can be approached by optical/ UV telescopes in space, but it is difficult to put big telescopes in space. On earth, the biggest 8-10 m optical/IR telescope approach this limit using adaptive optics to remove the blurring effect of the atmosphere.
At longer and shorter wavelengths there are different problems that stop this limit being completely reached. At radio wavelengths, huge diameters are needed to get even close to the angular resolutions of optical telescopes. This can only be achieved by linking radio telescopes together into arrays and using interferometry techniques - synthesising larger apertures. An excellent example is the mm-wave ALMA array that will soon be operational in Chile.
At short wavelengths, there are no effective lenses to focus the light. X-ray telescopes must use nests of grazing incidence mirrors. Whilst these achieve resolutions that can now approach those of ground-based optical telescopes (e.g. Chandra), the compromise is limited collecting area. Simple lack of photons can also affect the ability to resolve details in this part of the spectrum. At gamma wavelengths this is even more the case. Ingenious solutions include "coded masks" to perform imaging, but it is not possible to get anywhere near the fundamental resolution limit.
Your question is very (too) broad and I have limited my answer to an astronomical perspective. Apologies if this not what you were looking for.