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The latticework structure known as the cosmic web provides a valuable insight into the assembly history of large-scale structures. Despite the variety of methods to identify the cosmic web structures, they mostly rely on the assumption that galaxies are embedded in a Euclidean geometric space. Here, we present a novel cosmic web identifier called sconce (S pherical and CON ic C osmic w E b finder) that inherently considers the 2D (RA, DEC) spherical or the 3D (RA, DEC, z) conic geometry. The proposed algorithms in sconce generalize the well-known subspace constrained mean shift (scms) method and primarily address the predominant filament detection problem. They are intrinsic to the spherical/conic geometry and invariant to data rotations. We further test the efficacy of our method with an artificial cross-shaped filament example and apply it to the SDSS galaxy catalogue, revealing that the 2D spherical version of our algorithms is robust even in regions of high declination. Finally, using N -body simulations from Illustris, we show that the 3D conic version of our algorithms is more robust in detecting filaments than the standard scms method under the redshift distortions caused by the peculiar velocities of haloes. Our cosmic web finder is packaged in python as sconce-scms and has been made publicly available. [ABSTRACT FROM AUTHOR]