I've listened to clearance delivery feeds on LiveATC. Is it possible for clearance delivery to assign VOR or GPS-A approaches instead of standard terminal arrivals (STARs)? Or would they just give you routing to a VOR or another waypoint near the arrival airport, and say "direct" afterwards to indicate that the TRACON for the arrival airport will assign the specific approach?
No. Normally you don't know which approach you are going to be using at your destination because you can't be sure which runway will be in use. The STARs at larger airports are to provide standard routings for all/most the of airport's approaches/runways and each STAR is based on a different arrival direction, so the STAR in your clearance will usually be based on where you're coming from and you usually include one in your filed flight plan, that you've picked because its initial waypoint links in nicely to your route. The actual routing within the STAR you will be using, you find out when you listen in on the destination ATIS to find out the approach in use as you plan your arrival.
Once you get there, you may only fly one or two legs of the STAR before ATC gives you a heading and now you are on radar vectors to get lined up for the approach and you just do what you're told and the rest of the STAR routing goes out the window.
The key thing in IFR is accommodation of communication failures. Comm failure procedures are intended to allow you to complete your flight by following your most current clearance and observing SOPs that allow ATC to predict where you're going to go and when, with reasonable confidence, so that they can work traffic around you. If you've declared a comm failure (7600) enroute, you would fly the STAR routing and altitudes as published and use as much common sense as possible.
For example, a lot of STAR routes kind of drop you off at a Downwind Termination Waypoint for a particular runway and indicate radar vectors beyond that. In a comm failure situation you are expected to fly the published route to the DTW and from there you are expected to find your own way to the approach by the most logical, predictable routing that causes minimum disruption.
If the destination has no STAR, your clearance will be to the last waypoint in your route that gets you close to the airport, and that will be your clearance limit and from there you would get radar vectors to line up with whatever approach is in use. In a case of a comm failure, you are supposed to proceed to your clearance limit, hold there if you're early, then proceed to fly the appropriate approach at the time specified in your flight plan by the most logical routing, so that ATC can predict where you're going and when.