Is there a probability distribution for which the standard deviation is proportional to the mean?
For the standard deviation to be proportional to the mean, the mean must be able to vary, in which case we're really talking about a family of distributions. [Edit: oh, I see whuber makes the same point above.]
Such a family is sometimes said to have constant coefficient of variation.
There are many. Two common examples:
(i) The lognormal distribution with constant $\sigma$
(ii) The gamma distribution with constant shape parameter. This includes the exponential as a special case
I have been searching the web for a nice table comparing the properties of probability distributions (including mean and std).
Some books contain summary information like this.
Wikipedia doesn't have many distributions in one table, but it does have summary information on many distributions.
It's possible to fund some such tables on-line. Here is one example.