Too add another perspective:
It's all a matter of compromise. A chip with both analog and digital functions is considered a hybrid IC. The Atmel uC is a good example of this, since it does have analog inputs. As dc42 said, it has a dedicated analog ground pin for this reason. How much separation you provide is entirely up to you and the needs of your application.
For ultra-precise 24-bit ADC, for example, you would want to go to great lengths to ensure the supplies (Vcc and Gnd) were as stable as physically possible. This means heavily filtered Vcc, and super low-impedance grounds. The thinking behind this is that there's no such thing as a perfect ground. All traces, wires, pins, and components have some impedance. The longer and more convoluted the path, the more impedance. The more stuff sharing that path, the more that impedance starts to manifest as current. In other words, Ground ceases to be 0.0000v, and instead ends up being 0.01v, for example. Worse still is it will fluctuate as loads change. This is exactly what you don't want with a "reference".
So, regarding whether it's a good idea to use separate regulators and whatnot, well... again... it depends. You could power every component with its own battery. You could share the battery and provide separate regulators. You could share a regulator and put local filtering caps right up at the Vcc pins of the ICs. It depends on how much, in practical use, one component affects others. Bear in mind, you may get better results from using filters (inductors, caps, resistors, and diodes in various arrangements) where the Vcc line splits than you get with anything short of dedicated power supplies for each IC.
For grounds, you don't want any filtering. Those should be as direct as they can be back to the supply ground terminal. The idea of separation here is that a given trace or wire can only funnel so much current before the voltage on that line is non-zero. (Technically, it can't funnel any current without being non-zero. Really I mean "with a negligible increase.") Digital circuits tend to switch between fully-on and fully-off. This causes power dips on the off-to-on transition, and power spikes on the on-to-off transition. The filtering caps will absorb this and shunt the resulting AC noise to ground. So, you don't want your analog ground reference twiddling around with spikey transients when it should be a perfect flat 0.000v. The solution is to provide a dedicated, clean ground.
You may be thinking, "well doesn't the ground at the supply have all this accumulated mess on it?" Yes, yes it does. BUT, the important thing to remember is that "Ground" is not a concrete, tangible thing. It's a reference point. Vcc (and any signal) is relative to this reference. If Vcc is +5 (with respect to ground), it doesn't matter if ground is really +60 with respect to some other arbitrary circuit. The difference between Vcc and Ground is all that matters to the component using that supply ***. As long as all components share the same precise reference point, their relative offsets are all perfectly aligned.
Once you grok this, you'll find yourself in the inevitable debate: Which is better -- star grounding, or ground planes? Many experts will argue one side or the other. The truth is, star grounds have the advantage of every spoke of that star being at the same exact reference potential. But, a ground plane implies lots of copper area on a PCB, and therefore minimal impedance. So which is better? Dedicated ground planes (for each device) in a star topology of course! But then you run out of PCB area... So you must compromise and decide which components can share a plane without dirtying it up too much, and isolate those that would poison the well for everything else.
*** Of course, this assumes that your circuit is not directly (DC) coupled to some other circuit where Ground is +60. This is where isolation comes into play. For instance, if two circuits are isolated from mains through a transformer, they're both floating with respect to each other. Putting a probe of a meter on each circuit's ground will have undefined results, since they have nothing in common. This is why you must always tie all connected circuit grounds together, so their voltages are all relative to the same reference.