There used to be big, powerful single machine supercomputers like Cray. Now it seems that mostly supercomputing is done on big clusters of Linux machines. What is the reason for this change? For example, were there problems in getting the single machines to scale well or shed enough heat from the processors? Are there any modern non-cluster supercomputers anymore?
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Yes, several non-distributed systems are still in operation. You can check out the TOP500 list. Note that although they are not distributed, they remain massively parallel machines, always made of thousands of processors divided in nodes.
Capacity: A reason for the popularity of clusters is the type of work needed. Many modern business problems are easy to divide into separate chunks of works that can be completed separately and re-assembled later on (embarrassingly parallel problems). In those cases, the price ratio of distributed systems is much lower, they don't require much research and development, and the performance is interesting (even though they generally don't match the top supercomputers - the K Computer is estimated to be equivalent to a million PCs).
Capability, however, relates to single large problems that are harder to distribute. They are often complex models where data need to be available at every step in order to solve. A good examples are scientific models like weather systems: you can't outsource the "cloud" calculations and the "wind" calculations to networked machines efficiently. Supercomputers remain important for these purposes.
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