Tourist Visa (L) will allow visitors to travel freely in most parts of China as tourists. (If your situation is not described by one of the other visa types, then this should be the one you need.)
More info: http://www.travelchinaguide.com/embassy/visa/tourist.htm
72-hour visa free transit:
Passport holders of 51 countries and regions can be granted a visa-free stay of up to 72 hours while in transit
Visa exemption (Singapore, Brunei, Japan):
Ordinary passport holders of Singapore, Brunei and Japan are allowed to enter China without visas for up to 15 days for tourism, including visiting family or friends, or minor business purposes provided they enter through nominated international ports.
For nationals of these three countries, a visa is required under following circumstance:
(1) Ordinary passport holder visits for tourism, business, visiting friends and relatives and stay for more than 15 days.
(2) Ordinary passport holder visits for the purposes of study, work, official visit, settling down and interview.
(3) Diplomatic and Official passport holder.
More info: http://www.travelchinaguide.com/embassy/visa/intro2.htm