What is the usual role of a faculty senate? Does it typically have any real power, or is it a rubber-stamp for the administrators? Although the presence of a senate suggests that universities nominally are governed somewhat by their faculty, it is frequently stated that there has been a trend towards greater managerialism, which has perhaps undermined the powers of faculty regarding university governance.

What current powers does a faculty senate typically have, or is it mostly a rubber-stamp for the administrators?
This question seems a bit wide to answer.  Are you interested in public or private schools?  Junior college or peer 1 research level?  US or elsewhere?  Let me answer from the prospective of a 4-year US state institution.  Generally senates seem to be what the faculty make of them, they can choose to become a rubber stamp, or choose to act...  

A. The senate functions as a voice of the faculty

By filtering and focusing the concerns of the faculty, the senate functions as a key voice of the faculty.  A voice selected or elected by departments across the campus. The senate can conduct dialog with the administration, offer press releases, lobby with legislators, join in with other institutions to tackle larger issues... Whether or not they are listened to, faculty member should speak together often about the central issues at the institution.   

B. Determination of General Educational Policy

Traditionally the senate allows the different disciplines to discuss joint concerns, such as the general education policy, starting new programs, ...  They are rarely the final voice (or even the key voice) in these issues, but they are an important one.  Anytime an educational goal has been set, it is the faculty who should determine the curriculum, these decisions usually pass through a senate body.

C. Maintaining the Faculty "handbook"

This may be given different names at different institutions, but the subset of policies and procedures related to tenure, promotion, post tenure review, usually fall under faculty review. Core parts are usually decided by the institutions board, but many of the details of faculty evaluations, ... are often decided by the faculty.  Even when they are not, the Senate provides a united voice to address these issues.

D. Protect the Faculty's Interests

For example, the Senate should (to the best of their ability) be involved in suggestion priorities for budgeting (whether listened to or not); in suggesting standards for academic freedom (and explaining to its member what it is, and what it is not); when necessary, the Senate should be ready to stand up for individual faculty when the institutions polices have been incorrectly applied;  ...