It is common for an IoC container to allow you to specify the lifestyle of an object. SimpleInjector, for example, provides Transient and Singleton out of the box (and also allows the user to create a custom lifestyle class). With SimpleInjector it's as simple as:
container.Register<ISvc, Impl>(Lifestyle.Singleton);
or
container.Register<ISvc, Impl>(Lifestyle.Transient);
No need for polluting your classes with composition logic.
As for configuration primitives, a common practice is to put those behind an interface. I put my connection string behind an IDatabaseConfiguration interface, for example, and its implementation reads the value from web.config. I then inject that interface into my data services.
EDIT: Disregard above. Keeping it in so that comments make sense.
The following doesn't actually answer the question, but it may solve the problem that prompted the question. You said that the class does some useful caching, which clues me in on that it is taking on an additional responsibility. I would therefore recommend that you do not attempt to register that single implementation as a singleton, but instead create a decorator class around that implementation.
container.RegisterDecorator(
typeof(ICommandHandler<>),
typeof(CommandHandlerCacheDecorator<>),
Lifestyle.Transient,
x => { return /*some logic that looks for and
finds that one class you want
to decorate*/
});
The decorator needs to be a transient, because it will have a transient reference to the command it decorates. However, it could then access a separate singleton class (via dependency injection of course) or cache object that handles the actual caching.
EDIT: Examples to single out the one class to decorate. There are plenty of options depending on just how specific you want to be. Again this is with SimpleInjector, but I'm sure other containers have their own analogues.
//decorate a specific class
x => { return x.ImplementationType.FullName == "My.Commands.Web.SomeName"; }
//decorate all classes that share a certain namespace
x => { return x.ImplementationType.Namespace.EndsWith("Commands.Web"); }
//decorate all classes that implement the same interface
x => { return x.ImplementationType.IsInstanceOfType(typeof(ICouldCache)); }