I know that evolution can happen quite rapidly in single celled organisms. How about animals? Has there been any record of new speciations over the past 1000 years? In this video it is claimed that Darwin's finches arrived at Galápagos Islands just a few hundred years ago.
The video is a simplification of the speciation concept in order to convey principles of evolution using Darwin's finches as an example. Very rapid speciation (obvious speciation in 1000 years or less) for animals with much longer lifespans, is probably not documented. However, much of a potential answer to your question depends on how you define "speciations". Species is a very fluid term. When speciation is defined as major genetic changes resulting in obvious differences that distinguish two different populations, it is easier to say "yes" it can occur in shorter periods of time. For those differences to accumulate and increase to the point where animal phenotypes and genetic differences make reproduction (genetic exchange) completely impossible probably takes much longer. Here are two links that I found helpful.
The speed of evolution is controlled by three things how fast the organisms reproduces (aka how short the generation time is)(faster/shorter = faster), how large the population is (smaller = faster), and how strong the selective pressure is (stronger = faster).
If you want a list of observed speciation events you can start here and a more indepth look and a longer list of speciation here. However a list of all know examples would be very large, outside the size available for posting here.
As for the finches they arrived on the island about 2.3 million years ago based on genetic evidence, some species on the islands may only be a few hundred years old but that is very different then when the first finches arrived on the islands.