What are the evidence that all life today descended from a common ancestor (LUCA), and which organisms (if any) challenge the concept?
It is quite likely (but impossible to study) that the first stages of life originated more than once. However, the evidence that all currently existing life originated from the same population is very strong. 

On the most basic level, many of the basic building blocks of life, RNA, DNA, amino acids are chiral, which means they come in multiple forms, but for the basic processes in nature, always the same one is chosen. For example, the DNA double-helix is almost always right-handed in nature,
see also here. If there were multiple origins of archaea and bacteria, we would expect some to pick another form, if not other molecules. In addition, many crucial proteins such as DNA- and RNA- polymerases are very similar between all species, the probability that they would have originated independently is simply minuscule.