How useful are quizzes in undergraduate math courses?

I have implemented a variety of different quiz schemes over the years:

1. I give one quiz paired with each test. This is often too long for class, but I give it anyway as a form of test review. This was very popular with the students, but it eats a lot of time. It also forces me to be done with tested material at least 2 meetings before the Test itself. It was not always possible to get them feedback in this mode since the quiz was often about 3 days before the actual test.
2. I gave a short quiz at the start of every lecture where they were able to copy down the solution to one of the assigned textbook problems. The trouble with this was trading back and forth 40 assignments gets old in a hurry with 30-40 students. The downside, athletes always have an excuse and somebody's going to get married etc... there is a big time stamp for interfering with folks schedules here. So, some strategy to deal with missed quizzes equitably must be found. My solution was to just award them the grade of the quiz before or after the missed quiz. In addition, vonbrand is correct, we did have a problem with a student just copying down solutions from the online solutions manual brought to us by google. I will say, the student was good and showed understanding on tests, so perhaps that was research more then outright cheating. It is a problem though, so to do this right you really need to write your own quizzes.
3. I give a weekly quiz. This I usually do in a junior level course. Here I have a weekly homework (which I write most of the problems for to avoid the google problem ) and the other day of the Tuesday/Thursday schedule is a quiz based on the homework and/or lectures from the previous week.
4. just give tests and homework (no quizzes) this I do either when my schedule is too cramped to deal with quizzes, or when it is a senior or special topics type course.

Only option 2 is really intended to get them working in the text ahead of lecture. I think this confused them a bit because of the perceived unfairness of the method: how can I test them on what I haven't even taught yet? Thus, to ward off this (understandable) complaint you must be careful to only give quizzes which are answered on the basis of a merely cursory understanding of the material as it would arise from a first reading.

Long story short, I'm tired of attempting 2. at the moment. My current strategy to get them to read for the course is to put an explicit problem on each homework which has them affirm they have read certain sections of the book and my notes. I think this is a bit more polite and communicates my expectations for them repeated in the semester. I can write a sentence in the syllabus about reading before class, but it's better to remind them often of their duty as a student; to study.