While traditional, flying-spot, spectral domain OCT systems can achieve MHz linerates, they are limited by the need for mechanical scanning to produce a B-mode image. Line-field OCT (LF OCT) removes the need for mechanical scanning by simultaneously recording all A-lines on a 2D CMOS sensor. Our LF OCT system operates at the highest A-line rate of any spectral domain (SD) LF OCT system reported to date (1,024,000 A-lines/s). This is comparable with the fastest flying-spot SDOCT system reported. Additionally, all OCT systems face a tradeoff between imaging speed and sensitivity. Long exposure times improve sensitivity but can lead to undesirable motion artifacts. LF OCT has the potential to relax this tradeoff between sensitivity and imaging speed because all A-lines are exposed during the entire frame acquisition time. However, this advantage has not yet been realized due to the loss of power-per-A-line by spreading the illumination light across all A-lines on the sample. Here we use a supercontinuum source to illuminate the sample with 500mW of light in the 605-950 nm wavelength band, effectively providing 480 µW of power-per-A-line, with axial and lateral resolutions of 1.8 µm and 14 µm, respectively. With this system we achieve the highest reported sensitivity (113 dB) of any LF OCT system. We then demonstrate the capability of this system by capturing the rapidly beating cilia of human bronchial-epithelial cells in vitro. The combination of high speed and high sensitivity offered by supercontinuum-based LF SD OCT offers new opportunities for studying cell and tissue dynamics.