In this thread I'm going to go over the most common rolls for given encounters and common results for certain conditions. This might be considered spoilers or meta-gaming by some, so if that bothers you, for your own enjoyment back out now.
Generic Encounters
Generic encounter rolls are mostly pretty spread out, but the most common roll in Cities will be Influence. By contrast Wilderness encounters are never Influence; Observation rolls are slightly more common than any other in Wilderness encounters. Lore is slightly more common in Sea Encounters, but any could pop up. Notably Sea encounters contain the few encounters where bad things happen with no roll; if you draw the card, you're hosed.
Specific Cities
Arkham - Most common skill roll is Lore
San Francisco - Most common skill roll is Will
Buenos Aires - Most common skill roll is Lore
London- Most common skill roll is Observation
Rome: Most common skill roll is Influence, with Will a close second
Istanbul: Most common skill roll is Will
Shanghai: Most common roll splits between Influence and Observation
Tokyo: The most common roll is Influence.
Sydney: The most common roll is Influence, with Will a close second.
Al cities except London have an encounter asking you to spend a clue token to gain the benefit for the encounter. London asks you to gain the Debt condition.
All cities have an encounter where you may become Delayed in exchange for greater than normal benefits.
All cities have an encounter where you may become Blessed rather than receiving the normal benefit from that city.
Remember that while you cannot look in the encounter deck it is legal to look through the discarded city encounters to see what your chances are of getting a specific encounter.
Expeditions
1 third of the expeditions, once per location, moves back the Doom track by one on a successful encounter. The other two thirds provide an Artifact.
1 third of the expeditions, once per location, are Observation rolls primary. The other two thirds are Strength rolls primary. Secondary rolls are most likely Lore, Observation and Will, in that order. Your best bet for a successful expedition is a character with high Strength and Lore. For failed rolls, Strength, Influence, and Observation are all equally common secondary rolls; Lore and Will are never rolled.
11 of the expeditions will provide their benefit if you pass the first roll. 7 require both rolls passed.
The most common pitfalls of Expeditions are Amnesia and Injury Conditions, particularly Leg Injuries.
Other Worlds
Most Other Worlds require Will as their primary roll, with Lore, Influence, and Observation coming up in that order. Strength is never the primary roll.
Lore is the most common secondary roll on a pass, with Will be the next most common. Successful Gate closing means high Lore/high Will characters.
Will and Strength are the most common secondary rolls for failing.
Six Gates, or one fourth of the deck, can be closed simply by passing the first roll.
Six Gates can be closed even if the primary roll is failed.
The remaining twelve require two successful rolls to close.