Estimating 30 seconds
There are a few instances that I can think of where you need to estimate 1 minute. All pilots need to know how to reverse course when inadvertently entering IMC. The procedure is to note your heading and then make a 180° standard-rate turn. This turn should take 1 minute if you do not make any power adjustments and you maintain altitude. The heading indicator is the primary means of determining when to roll out. Roll out 10° before you are at the 180° mark. If your HI is inoperative, you can use the compass to judge when to roll out. The compass will lag when on a northerly heading and lead on a southerly heading. There are no compass errors on an east or west heading. Another way is to estimate how long a minute takes. This link takes you to a clock that lets you practice 30 seconds.
Another instance where you need to estimate 60 seconds is when flying a holding pattern. Entering a holding pattern is a time of high workload and it is not inconceivable that you might forget to hit the timer, or that the timer might malfunction. Knowing how long 60 seconds is could come in handy.
This link takes you to a clock that lets you practice 10 seconds.