Today I was browsing through some questions on this site and I found a mention of an enum being used in singleton pattern about purported thread safety benefits
In my experience I have seen Enum usage sometimes cause systems to be very difficult to change. If you are using an Enum for a set of domain-specific values that change frequently, and it has a lot of other classes and components that depend on it, you might want to consider not using an Enum.
For example, a trading system that uses an Enum for markets/exchanges. There are a lot of markets out there and it's almost certain that there will be a lot of sub-systems that need to access this list of markets. Every time you want a new market to be added to your system, or if you want to remove a market, it's possible that everything under the sun will have to be rebuilt and released.
A better example would be something like a product category type. Let's say your software manages inventory for a department store. There are a lot of product categories, and many reasons why this list of categories could change. Managers may want to stock a new product line, get rid of other product lines, and possibly reorganize the categories from time to time. If you have to rebuild and redeploy all of your systems simply because users want to add a product category, then you've taken something that should be simple and fast (adding a category) and made it very difficult and slow.
Bottom line, Enums are good if the data you are representing is very static over time and has a limited number of dependencies. But if the data changes a lot and has a lot of dependencies, then you need something dynamic that isn't checked at compile time (like a database table).