When your opponent kakaris a stone at either 4-4 or 3-4 with a one-space approach and no other stones in the local area, is it ever correct to respond with either a two-space or three-space pincer, or is the one-space pincer the only one played? Under what circumstances (if any) is a wider pincer correct, and what would be the purpose of the wider pincer?
jwaytogo: Although there may be no stones in the local area, the stones in a global context affect the choice of a pincer. Here are some ideas to consider when selecting a pincer for a kakari on a 3-4 point.
If a pincer is an extension as well, then the distance of the pincer is not so important. It becomes a matter of preference. Note that tight pincers (ie. one-space) put a lot of pressure on the pincered stone, and tend to immediately induce a fight. A loose pincer (ie. three-space), is not as urgent, and focuses more on the extension aspect than the pincer aspect. Since a loose pincer is not as severe, your opponent may opt to tenuki. Tenuki is usually not an option in response to a tight pincer, although it can happen. If you want to start a fight immediately, then pincer tight. If you want to play it safe and develop with an extension, then pincer loose.
If a pincer is not an extension, but a play into your opponent's sphere of influence, then it is more common to pincer loose, as this will give you more room to maneuver.
In response to 4-4 points, you can usually pick a pincer due to a certain global strategy. Like when you play sanrensei, and your opponent keima approaches a 4-4, a low one-space pincer or a high two-space pincer can potentially enable you to build a wall facing your sanrensei formation, leading to a good global strategy.
Although I am typing this from experience, a lot of these rules are probably taken from Yilun Yang 7p's book on the Fundamental Principles of Go. He also gives rules on how to pick a high pincer vs. a low pincer. For fear of violating copyright (cannot tell what is my knowledge and what is knowledge I gleaned from the book...), I should probably not elaborate too much. You might want to consider buying the book...