"Shall I" is common, in my experience, in Britain. It's used for a polite offer and means the same as "Do you want me to".
Here are some natural situations:
Shall I meet you at 8pm
Shall I open the present for you?
Shall I go first?
Shall I be mother? (funny expression for when you offer to pour the tea / coffee, or serve food to others during meals)
Shall I do the honours?
"Should I" = "Is it a good idea for me to..."
e.g. Should I climb Mount Fuji in December or not?
Should I speak to you later (if you are busy now)?