What causes someone to frequently see particular patterns of digits when they look at a clock? Over the past few years when I randomly choose to look at a clock I typically see particular patterns of times. The main pattern is repeating digits (e.g., 04:04; 11:11; 16:16). This occurs on multiple clock-based devices.

What can explain the tendency to frequently see particular patterns of time when checking the time?
In general humans are excellent at seeing patterns in randomness. 
It seems like you have attached special meaning to certain patterns. It seems similar to the way people form superstitions. 

Checking the time may also be something that we do so regularly and automatically that we do not know how often we do it. And it may be that the pattern confirming cases rise up and appear to be of significance, whereas the many other non-conforming patterns go unnoticed. 

We also have a fairly reasonable internal clock. And people also form habits. So it may also be that you have habituated to checking the time more frequently around these times.