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The main issue regarding signing statements is the administration's claims of authority and his attempts to "unbalance" power.
Each of the problems that are the result of the administration's claims of authority - warrantless surveillance, torture, waging war, etc. - need to be dealt with individually by Congress. It's not going to be quick and easy. (I'm not a lawyer, btw.)
A CRS Report dated 9/26/06 explains the significance of signing statements. This is from the conclusion:
While the broad assertions of executive authority contained in these statements carry significant implications, both practical and constitutional, for the traditional relationship between the Executive Branch and Congress, they do not have legal force or effect, and have not been utilized to effect the formal nullification of laws. Instead, it appears that recent administrations, as made apparent by the voluminous challenges lodged by President George W. Bush, have employed these instruments in an attempt to leverage power and control away from Congress by establishing these broad assertions of authority as a constitutional norm.
It can be argued that the appropriate focus of congressional concern should center not on the issuance of signing statements themselves, but on the broad assertions of presidential authority forwarded by Presidents and the substantive actions taken to establish that authority. Accordingly, a robust oversight regime focusing on substantive executive action, as opposed to the vague and generalized assertions of authority typical of signing statements, might allow Congress in turn to more effectively assert its constitutional prerogatives and ensure compliance with its enactments. http://www.fas.org/sgp/crs/natsec/RL33667.pdf