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 don't 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, i.e. 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:
Helmet mounted mirror are typically affixed with an adhesive pad or with a clamp to the front-side of your helmet. Typically they will stick out a little further than other head mounted mirrors, which is a good thing. It allows you to more easily see around your own head and you do not have to look off to as extreme of an angle. For adhesive type mounts the downsides are that the adhesive mounting point is (or is supposed to be) permanent and can potentially damage the helmet if removed. For some helmets it can be difficult to find an adequately flat surface to attach the adhesive pad to. The adhesive mount, if not appropriately attached, can fall off. If properly attached the mounting point is, in theory, permanent, although typically the mirror and the arm are removable from most mounts should you not want to use the mirror. Some of the same types of problems can arise from the clamp style helmet mount mirrors. The clamp has the potential to damage the helmet- though it is unlikely if you use due caution- and there may not be an appropriate mounting point for many helmets. If not adequately attached, the clamp may loosen and fall off.
Glasses frame mount
Glasses frame mount mirrors are similar to helmet mount mirrors in many ways, but typically sit closer to your face. This can make it a little harder to see around your own head or cause you to have to view the mirror at a more extreme angle. Mounting is easy and they are not permanent. On the other hand, they will not fit all styles of glasses and many of the plastic models are prone to breaking. Metal framed models are available which are less prone to breakage.
Glasses lens mount
Lens mount mirrors adhere to the inside of your glasses. They look great because onlookers can't tell that you are using a cycling mirror, and mirrors have a bit of a stigma attached to them, albeit undeserved. The downside to these mirrors are that depending on the shape of your glasses and the shape of your head, you might not be able to see anything with them besides your own face. It's also worth pointing out that if your require corrective lenses (not contacts) then this option is out by default. If this style of mirror works for you, you are likely in a fortunate minority.
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)
These mirrors have a plug that fits into the end of your handlebar to secure it in place. Many sit up higher than the bar itself, though some models are inline for a cleaner look and less obstruction of the hand(s). These mirrors are typically the easiest to view of the bike mounted mirrors due to their height and the distance out from the bike and rider's body. The other edge to the blade is that they are likely the easiest to knock into something, and depending on their shape they can get in the way. Some models will fit both flat bars and drop bars.
Bar end mount (drop bar)
These mirrors work the same way bar end flat bar mirrors work and some models fit both bar styles. For the inline style as pictured above, you get some tradeoffs. The mirror is less obstructive and a little less likely to get bumped into things, but to view it you do have to take your eyes off the the road in front of you more so than with a mirror that sits up higher. Due to the fact that drop bars are narrower than typical flat bars you may end up looking at your leg or bike, especially when turning in the same direction as the side of the bike the mirror is mounted on.
Hood mount (drop bar)
There are a couple of different mounting options for this type of mirror- one for integrated shift/brake levers, and one for non integrated levers. Both manage to bring the mirror up higher and further away from the bike, giving you some of the advantages of a raised flat bar style mirror. For the non integrated style lever, these can be a good option if you have bar end shifters. The mirror will still be a little tighter in on the bike, and both types of mounting options run the risk of damaging the lever in the event of an impact.
Handlebar mount (clamp/strap style)
These typically have a plastic clamp, or a rubber or velcro strap to affix them to a handlebar. They are an option if you have no other available mounting points on your handlebars. On the other hand, chances are if all other spots are taken, you will have a hard time finding a place where this type of mirror won't be in your way.
This is a somewhat unusual option that mounts directly to the frame of the bike. It will always look directly behind your direction of travel, regardless of where the bars are pointed, and it obviously won't take up any room on your handlebars. This may be a good option for time trial and triathlon bikes where mounting options on the bars are virtually nonexistent. The biggest problem with this kind of mirror is that it is going to be very difficult to see around your legs as it sits in so tight to the bike. I have not used this type of mirror, but I believe there could be a slight risk with some configurations of knocking against the mirror when out of the saddle and climbing steep ascents.
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
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.
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.