Choosing Resistor Values
Selecting a resistor depends on the circuit/block in question. As you already mentioned, different applications demand different requirements.  

For example, here are a few things to consider:

Pull up / down

Small resistors (aka "strong" pull up/down) are used when some nodes must be held at a given state "regardless" of the other impedances connected to the same net. However, in most cases, it leads to higher idle current consumption. Furthermore, due to their low resistive path, they pull the nodes faster to a given state.

i2C

The resistors used to pull up the bus line have a direct impact on the data transfer speed which can be used, and also how far apart the master and slave devices can located.

Bleeder resistor

Some circuits require a passive discharging of capacitive elements. A good example is a AC/DC power supply which should have its input capacitors (connected to the mains) discharged upon removing it from the mains, in order to prevent exposure of energized pins.  

Voltage supply regulation

Regulated voltage supplies rely on feedbacks to have their output voltage controlled. Most of the time this is accomplished with a resistive voltage divider. Due to the limited amount of resistor values, selecting a combination of resistors which lead to the required output voltage can be challenging. Here you can usually find such resistors \$22.6k, 11.5k\$ etc.

Compensation Network
Compensating loops, such as those of oamps, require placing poles and zeros at specific locations. As mentioned before, the limited amount of resistor and capacitor values may require to pick unique combinations in order to guarantee stability.

Note: These are just a few examples. The number of  applications are countless.