It's all about the re-oil procedures. If everyone is getting fresh stuff, it barely matters and comes down basically to what time of the day you bowl best. In a re-oil situation, I would prefer second shift (I know what I have to shoot, it's not first thing in the morning, etc.). If there's no re-oil, get first shift, unless the pattern is so heavy and/or flat that you'd benefit from having someone in front of you to break it down.
If it's like what the above poster mentioned and they're just adding oil to the heads, what you do in that situation is go find the tournament director and drop your ball on his foot. Such a scenario means the tournament does not offer equal shots for all bowlers and that's not fair.
Jess