This paper constitutes the first part of a study that addresses three rich historical themes in harmonic analysis that, in the non-zero curvature setting, investigate the boundedness properties along variable curves for(I) the linear Hilbert transform and analogous maximal operator,(II) the Carleson-type operators, and(III) the bilinear Hilbert transform and analogous maximal operator.Our Main Theorem in this paper states that, given a general variable curve γ(x,t) in the plane that is assumed only to be measurable in x and to satisfy suitable non-zero curvature in t and extra (mild) non-degeneracy conditions, the operators at (I) and (II) defined along the curve γ are Lp-bounded for 1<p<∞.Our result provides a new and unified treatment of the first two themes as well as a unitary approach within the x-variable setting for both the singular integral and the maximal operator versions of (I).At the heart of our approach is the LGC–methodology introduced here, which encompasses three key ingredients:(1)phase/space discretization that confines the phase of the multiplier to oscillate at the linear level,(2)adapted Gabor frame discretization of the input function(s), and(3)cancelation extraction via the analysis of the time-frequency correlation(s) capturing the non-zero curvature of γ.