You can see city lights from orbit on the night side of Earth. A lot of science fiction (yes, sorry... it's one of those questions) has dialogue such as "on a clear night you can see XYZ city on the Moon."
Obviously, we have no cities on the Moon, but we do have cities on the Earth. Did the Apollo astronauts see them from the moon or at least lunar orbit?
It appears the answer is "no".
Apollo mission reports describe night-time lights from Earth orbit, but sightings at lunar distances are notably absent. For example,
The sights in earth orbit were spectacular; even on the dark side, where thunderstorms and fires in Africa captured the crew's attention. The earth-orbit timeline provided sufficient time for viewing the earth, for unhurried crew adaptation to zero-g, and coping with hardware problems.
But these are in Earth orbit. The same mission reports say nothing about the dark side of the Earth at lunar distances.
There are too many bright objects, including the Sun, Earth, Moon, and (on the way there) the lunar module, that make seeing dim objects difficult. Sometimes they could not see most stars, even with the help of the on-board telescope:
Commencing with the Apollo 7 mission, all crews reported that several minutes were required for the eyes to adapt to the recognition of constellations when the command module telescope was used at night in either earth orbit or lunar orbit and in earthshine light conditions. With the lunar module attached during translunar coast, sun reflections from the lunar module into the optics prevented any but the brightest stars from being seen with the telescope. During transearth coast, constellations could usually be recognized when the telescope was pointed away from the sun, earth, or moon. In general, to maintain platform alignment during translunar coast, the Command Module Pilot relied on automatic optics positioning to place reference stars in the field of view of the 28-power sextant.
Apollo Program Summary Report, p. 6-15
An eyepatch to help the CMP adapt one eye to darkness to use the telescope, and (on Apollo 16 and 17) sunglasses did not help.
The lenses of the sunglasses were not dark enough for comfortable use in lunar orbit. A significant amount of eye fatigue was experienced after looking out the window for extended periods, even with the glasses on.
Only Apollo 14 and 15 carried the slow-speed film (types 3414 and S0349) that can capture dim objects (see table 6-VIII).
Apollo 7 and 9 did photograph city lights. But these missions never left Earth orbit, and that's not what this question is about.
Apollo 8 took the famous "Earthrise" photograph. Half of the Earth is in sunlight, which obscures city lights on the dark side.
Once at the moon, the commander and lunar module pilot were more concerned with their landing on the moon, than trying to look back at Earth. Furthermore, the landings occurred during lunar daytime. The reflected sunlight is so bright that someone on the surface can't see stars, much less dim lights on Earth.
The command module pilot for Apollo 10-13 spent most of his time either taking pictures of the moon, or managing the spacecraft. Keep in mind that these were the shorter missions, and also that they needed the reconnaissance for the later missions.
Apollo 14 tried to photograph the dark side of the Earth, and failed.
Earth dark-side photographs were unusable because scenes were obscured by scattered light from the sextant optics, from sunlit areas of the earth, and perhaps from portions of the docked lunar module during translunar coast.
Apollo Program Summary Report, p. 3-88
Apollo 15 through 17 did not attempt to take pictures of the dark side of the Earth. Knowing that these were the last missions, the crew was kept as busy as possible.
Beyond LEO, once you're a few Earth radii away, far enough to see the entire planet, its nightside is a featureless black, at least to conventional cameras, in every one of the dozens of photos at http://www.planetary.org/explore/space-topics/earth/pics-of-earth-by-planetary-spacecraft.html, even the ones that show Earth as only a slim crescent.
Edit: As commented, these photos were exposed for the day side. Had they been exposed to reveal any night side detail, the day side would likely have been overexposed into solid white.
Handwaving and possibly erroneous math: The nighttime-glowing part of Chicago is about 400 km2, the same as a disc of radius 11 km. Compare nighttime Chicago to a 22 km diameter asteroid with a whoppingly generous albedo of 0.05. Then its absolute magnitude is about 12 (its magnitude seen from 1 AU away, in that table). Seen from 239,000 miles away instead of 93,000,000, 390 times closer, it's $390 \times 390 = 150000 = 2.5^{13}$ times brighter, so its apparent magnitude is $12-13 = -1$, as bright as Sirius.
My estimate is likely incorrect of what a 20 km asteroid's albedo must be to match nighttime Chicaglow (what terrestrial Midwestern astronomers nickname its light pollution). But at least, seen from the moon, Chicago is no brighter than Sirius, and probably closer to magnitude 3 or 4, barely visible even without a distracting magnitude -15 Earth visible.
An independent calculation. From the ISS, Venus is as bright as the city of Valencia at night. (Other ISS views of Venus had places on Earth that I didn't recognize.) Valencia's metropolis has about 2M people.
In the 1970's Earth's biggest megacity, greater Tokyo, had about 23M people. 11.5 times as many, so 11.5 times brighter. (Maybe less because lighting was less efficient then, maybe more because we didn't fuss so much about light pollution then.)
ISS is 400 km up, so about 800 km from Valencia in that image.
The moon is 385,000 km away, 480 times farther.
So from the moon, 1970's Tokyo was $11.5 / (480 \times 480) = 1/20000$ times as bright as Venus.
Venus has apparent magnitude $-4$.
$20000 = 2.5^{10.8}$.
So from the moon, Tokyo had magnitude $-4 + 10.8 = 6.8.$
Under optimum conditions, stars of magnitude 6.5 are the threshold of visibility with the naked eye.
Thus, with the magnitude -15 dayside earth and the magnitude -26 sun dazzling the astronauts (a factor that swamps all the other estimating errors in this answer), even the brightest nightside city was too dim for them to see.
Their only chance to see city lights would have been during a solar eclipse and even then, still only magnitude 6.8, if Tokyo was facing them, without clouds.
Here is another take at estimating the brightness of the city lights as seen from the Moon. As a starting point, let us try to estimate how much light is produced by the entire United States at night. (I would guess that this probably accounts for a significant percentage of the total light produced on Earth, and probably actually a majority of the total light seen if the Western hemisphere happens to be in view.)
This source claims that the annual energy consumption used in the U.S. for "commercial lighting" is 141 billion kWh. This, however, includes both streetlighting and lighting of commercial buildings; let us guess that half of this goes to streetlighting, so 70 billion kWh.
Next we need to convert this to power. We divide this figure by one year, and also account for the fact that lights are on only half the time (only at night); so we multiply by 2. (In reality, lights are probably somewhat brighter in the evening than in the wee hours, so for peak output we might want to multiply by a larger number - but I do not thing this effect is very significant.)
To convert power to brightness, we need to multiply by the luminous efficacy. The most common types of lamps used for streetlighting are LED and sodium lamps; both seem to have a luminous efficacy somewhere around 130 lumen/watt, according to Wikipedia.
To get the illuminance at the surface of the Moon, it remains to divide by $4\pi$ (number of square radians in a sphere) times the distance from the Earth to the Moon (roughly $400\ 000\ \text{km}$) squared.
Finally, to convert this to apparent magnitude, we use the formula given here.
Putting the pieces together, WolframAlpha gives a magnitude of about +0.7 — so among the top 20 brightest stars, but not among the top 10 (comparable e.g. to Betelgeuse or Aldebaran).
So this is definitely invisible during the day, swamped by sunshine and its reflection on the lunar landscape. Is it visible at night, next to sunlit crescent Earth?
I would go for "no". Here is a comparison point: the Earthlight, whose magnitude is somewhere in the -2 to -3 range (source), and which, from personal experience, is definitely visible, but not so easy to spot.
Now, the Earth seen from the Moon is about 50 times brighter than the Moon seen from the Earth (see this Q&A thread on Astronomy SE); while, if my estimate is correct, Earth's city lights seem to be about 100 times dimmer than the Earthlight on the Moon, as seen from Earth...
OK, you would say, but then maybe during an eclipse?
Well, still unlikely. Consider that during a lunar eclipse (which corresponds to a solar eclipse on the Moon), the Moon is still visible; so there is still a fair amount of light reaching its surface. Seen from the Moon, a solar eclipse looks like a dark Earth surrounded by the bright aureola of its atmosphere, backlit by the Sun.
How bright exactly is it? Well, according to this source (specifically Fig. 4), even in the deepest part of Earth's umbra, the illuminance is still no less than $10^{-6}$ of the usual illuminance. In other terms, the Earth's atmosphere is roughly $10^{-6}$ as bright as the Sun, which corresponds to a magnitude of about -11. This is probably still bright enough to swamp the faint glow of the city lights...
