There are several sites online offering manuals or advice for the Singer model 503a sewing machine. A site called ismacs.net offers a free downloadable PDF of the 503a Singer manual. Google 'ismacs.net Singer 503a' and you should find what you're looking for.
If you're looking for spare parts, tips on repairs, or just somewhere to discuss the Singer 503a model, then try tandtrepair.com, oldsewinggear.com or sewingpatternreview.com.
You could also try Singer themselves, on their customer helpline: 1-800-4-SINGER (1-800-474-6437), via email at their website, or by writing to Singer Sewing, 1224 Heil Quaker Blvd, PO Box 7017
LaVergne, Tennessee, 37086.
YouTube user Rayna Morgan has a series of films based around the Singer 503a, with lots of tips and useful advice:
