@maru This is important: this manual (and this threading) does not match your machine. The Singer 15 had several submodels, and a range of them 1) had a mirrored bobbin case, 2) reversed the direction in which the needle was inserted, and 3) reversed the direction in which the needle was threaded. You have one of those submodels. This manual is for one of the other submodels (with the configuration that became standard across multiple manufacturers of this design, and which Singer eventually returned to).

Here's a chart of the different submodels: https://web.archive.org/web/20240521105632/http://needlebar.org/main/15chart/

You need a manual for one of the ones with an 11 o'clock bobbin case (your machine is probably a 15-88, but the others will have the same threading; they have other differences that are not as relevant). You can find manuals for this and similar submodels at ISMACS: https://ismacs.net/singer_sewing_machine_company/p15.html