Most rares if they are going to spawn that day, will spawn at server up. Server up is right when the server comes back online from its scheduled maintenance cycle. Most maintenance cycles last about an hour. To be a little more precise the time it takes depends on the size of the shard. The smallest shard in UO (Population and Item count) is Siege, it can take only 35 minutes for Siege to get through its cycle, the largest should still be Atlantic, and it can easily take an hour and 15 minutes for its daily maintenance. In addition to the normal time your shard takes, there are also a lot of other variables. Even tho OSI will try and take it down at 5am and have it back up by 6am things come up that often change that time by 30 minutes or longer.
All American and European shards go down at 5am Local server time. All Asian shards go down at 8am Local server time. Oceania (and presumable any other Australian shards that might be developed) go down at 7am Sydney time.
This is still in limbo since the servers for the central shards were physically moved to an east coast location. Right now they are sticking to Central time, but might at some point go with the East coast time.
Siege Perilous
(Siege is scheduled to go down at 5am central time, but it actually goes down much closer to 4am Central time)