Regarding checks and balances the president, Congress, and the judiciary have on the bureaucracy, which ones are most effective in their wide use?
I think that all of the checks and balances elements are in play in the modern political setting. I think that we see a great deal of play between the legislative and executive branches in their articulation of limitations of each others' power. The executive branch's ability to veto legislation and the legislative's ability to kill any legislation that the executive might propose are both elements of a political game of "mutually assured destruction." There is much played out on this end because both the legislative and executive are seeking to appear in a certain way to the electorate. The checks and balances scenario can have different ramifications on perception and this might be one of the reasons why such debates are highlighted. This same interplay can be seen during legislative confirmations of executive appointments. In the ongoing drama between the legislative and executive branches, the checks and balances dance between them both is something that has a determinant impact on the bureaucracy and governmental affairs and how the body politic sees both.
First of all, please note that (in the way we usually use the term) the president does not have a check and balance relationship with the bureaucracy. Checks and balances are supposed to work between branches of the government. The bureaucracy is part of the president's own branch and so the president does not check or balance it.
The most effective check or balance on the bureaucracy is the legislative branch's power of oversight and budget. The legislature has the power to set the budget for any bureaucratic agency. It also has the power to summon officials from the agencies to appear at Congressional hearings to explain their actions (and often to be chastised by every member of the committee). These powers are used quite often as a way for the Congress (particularly when the President is of the other party) to affect what the bureaucracy can do.