What telescopes have observed anomalies in the light curve of Tabby's star, KIC 8462852? KIC 8462852, the F-star that inexplicably dims by up to 20% for short periods, is still getting alot of attention. A slow continuous dimming by about 0.3% a year has been identified recently. But all anomalies I know of come from the Kepler space telescope. It is an F3-star at 1500 ly distance. Shouldn't it be easy to observe its light curve with even relatively modestly sized ground based telescopes? Have such observations been done and have they confirmed any anomalies? You see, I suspect something weird went wrong with the Kepler telescope here.

EDIT: There have been studies of its brightness on archived historic photographs. But I'm more interested in observations simultaneous as Kepler or since then, to confirm either a gradual dimming or short deep dips.
I do not know whether complimentary observations of the light curve are being done, but I will try to answer the rest of the question.

Is it easy to observe the light curve with ground-based telescopes? No, it's quite difficult. From the Kepler homepage:


  Since transits only last a fraction of a day, Kepler must monitor all target stars continuously. Their brightnesses must be measured at least once every few hours.


It goes on to explain how the observed field was chosen - no downtime due to it being too close to the sun, no obstruction by asteroids, etc.

A ground-based telescope has the obvious problem that it can't observe during the day, so you need several telescopes in different parts of the world, and combine their measurements. Since the distortion patterns, viewing conditions, instrument characteristics etc. are all different, combining the data is anything but trivial if you want a precision measurement.

In short, a change in the point spread function between telescopes might introduce both systematic and statistical errors in your brightness measurement. Kepler is only one instrument and optically highly stable.

Is KIC 8462852 bright enough for a small ground-based telescope? Certainly. Its magnitude is 11.7, while Kepler wants to observe stars brighter than 14th magnitude (link above). This magnitude calculator estimates a limiting magnitude of 17 for a 1m mirror.

It takes time to set up an observation. Large telescopes like the VLT accept requests for observation time twice a year. After your observing proposal is accepted, it will still take weeks or months until your observation is actually made. For smaller telescopes, it might be easier and quicker, but it will take time to set up a monitoring programme, even if telescope time is available.

However, there are ground-based, long-term monitoring programmes. COSMOGRAIL is one that comes to mind. From their homepage:


  Obtaining regular observing time on telescopes in good sites was (and is still) not easy


and they've been going for a number of years by now.

So, it would be technically challenging but possible. Politically, it might not be so easy: Every observatory wants to make the best use of the observing time they have, and most are oversubscribed (more time asked for than available). You need a fairly solid science case to block several telescopes around the world for several years, but the result you'd get if all goes well is merely confirmation of someone else's result.