Getting the voyage
They will randomly appear at "Special Voyages" tab.
Only 6/12 adventurers have them:
Chef, Exile, Memory, Tengu, Trapper, Whaler
Adventurer have to go to specific island in the voyage's region to find a clue.
To receive a clue, the voyage must be successful and the island must be the correct one.
You can talk to the adventurer before you send the voyage, to point him to specific island (there is also convenient right-click option for this).
Adventurers keep track of which islands they been at, and hence will finally find the clue even if you do not interfere (provided you will keep sending them to clue voyages).
For each adventurer this is repeated 3 times, then you unlock his/her reward - repeatable special voyage.
Increasing your chances - Map tables
Each map table will give you increased chance of receiving clue voyage.
But it's more important feature, is that it tells you the correct islands to send adventurers to.
If any of the adventurers in port can receive clue voyage (it does not have to be currently in voyages list) you can "investigate" the map, and it'll highlight their islands.
If you have 2 adventurers that can get clue, then 2 islands will be highlighted without telling to who they belong, so you should keep some notes to decide what island belongs to whom.
Mind that single map will not work every day:
1 table = hint every 4th day
2 tables = every 3rd day
3 tables = every 2nd day
4 tables = every day
Also, if for example you have 1 table, and don't check on 4th day, you will not get hint on 5th day, but will have to wait till 8th day for it (another full cycle).
To get the rewards unlocked as soon as possible:
» Build 4 map tables (hint every day).
» Manage a list with adventurer/island mapping (to disambiguate hints).
» And do tell adventurers what island to go to before you send the voyage out
Provided courtesy of Ker Laeda, last updated 06 May 2015