First time venturing into the thread. I know next to nothing about umpiring so I'm going to answer this just based on my rationale.
- A bowler asks you to have a look at the ball, saying he is concerned about the seam. You take it from him but everything seems to be okay. As you are about to hand it back to him he mutters under his breath but audible to you only, that he thinks his captain may be involved in match-fixing. What do you do? (Assume an international match for this.)
Let the game go on for the day, keep my wits about me to see if I notice anything untoward, then report it to the Match Referee at the end of the session or at close (assuming he is the point of call for such issues).
- One of the true legendary batsmen of the game is playing his last evermatch innings. The match, regrettably, seems to be headed toward a draw. He has been playing well so far and has gotten to 82. He drives a ball but is caught at midwicket off of the inner part of the bat. As he starts to walk off the fielding captain runs up to him and has a few words before coming up to you. He says that he will withdraw his appeal and all further appeals and will let the batsman continue and in return the batsman will retire out when he has made a century. Can you allow this?
No, you got out fairly and squarely, only himself to blame. Off back to the pavilion with ya. As an umpire you have to put your foot down, it would cause too much controversy and the repercussions would be unforeseen. Best to avoid any of that.
- There is a very light drizzle of rain. It is not enough to stop play, so you allow play to proceed. But after 15 minutes, the drizzle still has not stopped. After 20 minutes, it has gotten even lighter but still persists. What do you do?
Good one, this is difficult.
Assuming the lighting conditions are fine, consult the groundsman to make sure he's happy that no damage is being done to his pitch, if so, play resumes. Not convinced with this answer though.