This question comes up frequently. It's clearly impossible for an optical telescope on the Earth to resolve any of the Apollo hardware on the Moon, since the best systems, using adaptive optics in the near-infrared, can resolve details of maybe 0.02 arcsec. A lunar lander of width 5 meters, at a distance of 382,000 km, subtends an angle of 0.003 arcsec. The Hubble Space Telescope isn't appreciably closer the Moon, and its best resolution is about 0.03 arcsec in the near-UV. Not good enough.
An infrared image taken by one of the the European Very Large Telescopes in
Chile, which has 8.2-meter mirror, shows some of the finest detail
observed from the Earth.
At
the ESO Press Release page ,
you can see images with an angular resolution of about 0.07 arcsec;
they show details as small as 130 meters across.
Still a lot larger than the lunar lander....
Is it possible that a lunar lander might cast a shadow long
enough to be detected from the Earth? Well, possibly --
but I doubt it very much.
We do have pictures of the shadows cast by
lunar landers, taken by several spacecraft which were
all in orbit around the Moon --
the Apollo command and service module (CSM),
the Clementine probe,
the Lunar Reconnaissance Orbiter (LRO).
as it orbited the Moon.
You can see for yourself that the shadows are very
inconspicuous.
In order for a ground-based telescope to see the shadows,
it would have to be able to take pictures showing details
as small as those visible in these pictures taken from lunar orbit.
I don't know of any such Earth-based pictures.
Alex R. Blackwell, of the University of Hawaii,
provided pointers to three photographs
taken by Apollo astronauts, in a posting to sci.astro
on Jan 3, 2001.
He wrote:
Apollo 16 site: High resolution vertical view
http://www.lpi.usra.edu/images/sapo/sapo_S30.gif
The caption reads "The Lunar Module casts a distinct shadow on the
Cayley Plains. The landing site is about 500 meters east of the rim of
Spook Crater, which was visited by the crew during the first EVA. (NASA
photograph AS16-4558[P])"
Apollo 17 site: Enlargement of high resolution view
http://www.lpi.usra.edu/images/sapo/sapo_S36.gif
The caption reads "The Lunar Module landed on the Taurus-Littrow valley
floor among several 100-meter-sized craters. The first EVA included
setting up the ALSEP station and an investigation of Steno Crater (the
largest crater on the lower margin). The second EVA included traverses
along the southern rim of Camelot Crater (large crater left of the
landing site). (NASA photograph AS17-2309[P])"
I would presume that these images are from the respective CSM SIM bay
panoramic cameras.
In an article
on space.com ,
written by Leonard David, 27 April 2001,
there is a picture taken by the Clementine lunar probe which
shows evidence for the Apollo 15 landing site.
The evidence was noticed by
Misha Kreslavsky,
of the Department of Geological Sciences at Brown University,
and
Yuri Shkuratov
of the Kharkov Astronomical Observatory in the Ukraine.
Last modified July 17, 2009, by Michael Richmond.
Pictures taken from Apollo modules
The only relevant imagery I could find on a NASA website of the Apollo
landing sites and hardware are the following:
Apollo 15 site: High resolution vertical view
http://www.lpi.usra.edu/images/sapo/sapo_S24.gif
The caption reads "The Lunar Module casts a distinct shadow on the mare
surface. (NASA photograph AS15-9377[P])"
Picture taken by Clementine orbiter