What options are available in terms of types of cycling mirrors? Where all can I mount them and what are the advantages and disadvantages to each?
Bicycle mirrors are going fall into two basic categories- the type that you mount somewhere on your bike and the type that you mount somewhere on your head. Both categories have their pros and cons, but many of them are subjective. A pro to one individual may be considered a con to the next. Within those two categories you have a variety of different options to choose from. Here's the breakdown of those that I am aware of.
On your head
One thing to keep in mind with head mounted options is that they are always in your field of vision. You may consider this a pro, or you may consider it a con. On one hand you dont have to look off of your course of travel by much to view the mirror, on the other hand it can be distracting to always have it in your field of vision and can be awkward to focus on. Head mounted mirrors always look directly behind where your head is pointed. That means that they're not always pointed opposite your direction of travel, ie the road and traffic behind you. Depending on how far away from your head the mirror is, it can be difficult to get the mirror adjusted in such a way that your head does not obstruct your rear view or to where you are not looking off to an extreme angle to view the mirror. Mounting can be finicky. On the plus side they tend to be very light weight and tend to stay adjusted for the duration of your ride, and they are generally less obstructive than most bike mounted mirrors. Your options for head mounted mirrors are:
Glasses frame mount
Glasses lens mount
On your bike
This category of mirror will be affixed somewhere to the front of your bike. They will not be in your field of view at all times, which depending on your preference may be mark for or against this category. You will have to take your eyes off from your course more so with a bike mounted mirror than with a head mounted mirror, and to varying degrees depending on where the mirror mounts. Bike mounted mirrors will be easier to look at since you can more easily focus on them with both eyes instead of largely just one as with head mounted mirrors. While riding in a straight line the mirror will always be pointed mostly behind you, but in turns they may point off to one side a little or a lot (the frame mounted mirror described below is the exception). Mounting points tend to be more secure than head mounted mirrors, however bike mounted mirrors have more of a propensity to stick out from the bike in some way. Hand in hand with that issue, bike mounted mirrors can be more of an obstruction than head mounted mirrors both in terms of getting in the way of your hands and in terms of getting knocked against things.
Bar end mount (flat bar)
Bar end mount (drop bar)
Hood mount (drop bar)
Handlebar mount (clamp/strap style)
Keep in mind that you may or may not find that any of these issues actually affect you. Preference is going to play a huge part in your decision. Most cycling mirrors are cheap so if you're unsure of what will work best for you, buy a couple of different styles and try them while you ride.
To add to jm2's post, there's also a mirror that is integrated into a bar:
Advantages: Foldable, unobtrusive, usage as bar
Disadvantage: Small mirror => very limited field of vision
The problem with nearly all bicycle mirrors is that they only allow viewing from one riding position. It would be a simple design to have a two-section mirror, such that you can adjust one section to see the reflection when seated, and the other section when standing, as when climbing steep hills.
The closest that I have been able to find are two bar-end mounted mirrors, adjusting them differentially. The main problem with this is that they are so far apart that the angle of view is quite different.
Beside different mounts, the mirror shape, size and convexity are also to be considered. Strongly convex ones might be good in city traffic as they give a wide field of view, but for seeing distant (but fast) cars on country roads a flat mirror is better.
Also consider that some mirrors may make the handlebar wider, which can be a problem when filtering in traffic or on narrow cycle paths or shared footpaths, but isn't an issue on country roads.
In practical terms, as everybody's bike, body, riding style and road environment is different, it can be very difficult to find a mirror or mount that works well for you. I tried literally dozens. I recommend, before buying a good expensive one, buy a cheap one in a 1-dollar/1-pound/1-euro shop and tape it to the handle bar in different positions to get a feel how much you can see and where the best mount point would be for you.
Besides bicycle mirrors, it's also worth looking at motorbike, e-bike, mobility scooter and wheelchair mirrors. I'm now using a motorbike bar-end mirror (aluminium frame with real glass mirror), which has the advantage that it's more solid than any bicycle mirror that I could find and doesn't wobble or fall off at potholes (of which there are plenty here), and also has better optical qualities.