How widely used are 印章 compared to signatures? 印章 appear all over chinese works, especially in the past. Most noticeably in documents relating to business and arts, companies and individuals would use seals in place of signatures. 

In modern China, how commonly used are these in opposition to western-style handwritten signatures? Celebrities obviously have handwritten autographs, and many institutions do still use 印章 to authorize or mark documents associated with them, but how likely is a common person to use one? In legal documents, do people require 印章 (as Japan does), or are written signatures accepted? 

Under which scenarios is each used, and how commonly are either seen in modern China (both the mainland and Taiwan, which I take are different in how much they still use seals).
Organizations often use 印章 while signatures are used personally.
If you are representing an organization, 印章 usually are required, but signature is acceptable for a person representing himself. 
Example:
A bank is providing a loan to me, on the contract there will be bank employee's signature, bank's 印章 and my signature, this is acceptable.
Of course, if you have a personal 印章, you can use it with your signature, that's fine.
PS: This is in mainland China, not sure the what the situation is in Taiwan