how to refactor many singletons

I have a medium-sized python program (~5000 lines of code), which I've built up over time, with no particular plan as I went ahead. The architecture I've ended up with consists of 5-6 large Singleton objects, each of which handle a certain aspect of my program, such as database comms, a web-server, data gathering client, and internal calculations.

I feel that using several singletons like this doesn't really make use of the true benefit of OO-programming, namely that it easily allows you to create numerous related objects. My singletons are also fairly dependent on each other, and if I grew the program a lot (say ~50 000 lines), I can see my approach becoming spaghetti code.

So I'm wondering if there is a better way of structuring my program. E.g. should my singletons actually be separate modules? But then how would I approach the attributes that are very neatly organised in my singleton objects? Are there other architecture options?

I'm not very familiar with design patterns (the GOF book is on my todo-list) so perhaps there are design and/or architectural patterns that would be better for my program?