1. Cyclists should pay "road tax"
Fake "Road Tax" payment tag (Source: ipayroadtax.com)
First of all, there is no such thing as "road tax" although many drivers would like to belive so. It is the excise tax what all drivers pay, even if they don't drive their car at all. That's because the excise tax is the tax of car ownership, not usage. Yet many drivers think that since cyclists don't pay for roads in a form of a "road tax" they should not use them. Roads are financed from income taxes, not "road tax", and those are paid by all drivers and cyclists.
2. Mandatory insurance
This has been discussed so many times yet still there is a large number of drivers believing that cyclists should pay a mandatory insurance. The reason why drivers have to be insured is that they use a 2 tonne vehicle that moves at high speed. Any accidents would likely result in injuries, even fatal, to cyclists or pedestrians, but less likely to drivers themselves. Cyclists on the other hand, operate a lightweight vehicle moving at low speed that is unlikely going to kill anyone, except maybe its rider. The damage done by bicycle in case of a collison is minimal compared to the one done by car and it doesn't justify the need for insurance. This is exactly why pedestrians are not insured either.
3. Bicycle license plates
Custom license plate (Source: http://chiccyclist.blogspot.com/2009/06/cambridge-bicycle-license.html)
Some drivers believe that since they have to display a license plate on their vehicles, cyclists should do the same. You know, it would help identify those cyclists who break the law running red lights. The problem is that license plates don't stop drivers from breaking the law so why would they stop cyclists? Actually, why don't we require pedestrians to wear license plates as well? They too disobey law by crossing the street on red. Bicycle license plates would not increase safety but only add cost of owning a bicycle.
4. Mandatory helmets
Must wear a helmet (Source: yehudamoon.com)
Well, this is not just an idea. In some countries it is the law. Requiring everyone to wear a helmet does not prevent accidents. It may result in less injuries. It definitely leads to decrease of number of cyclists. It discourages users of city bike sharing programs. In general, it leads nowhere.
5. Bikes off the streets
Some drivers would love to believe that cyclists don't belong to streets. Those are the same who think bikes should be only used for recreation in state parks. But bicycles are vehicles too. Just like cars, trucks and excavators. There is no good reason why bikes should not be allowed on streets. Except the highways. But the same way it is foolish to require cyclists to use bike lanes exclusively. Just like you don't expect drivers to use highways only.
6. Signal pothole-evasive maneuvers
That is the most interesting one I heard. Apparently, it aggravates drivers when cyclists ride in a some "zig-zac" pattern trying to avoid potholes and they think that cyclists should signal those maneuvers. Why should they, if they don't change lanes? When you drive on a country road that turns 45 degrees, would you signal that? Cyclists should remember that they move between much heavier and faster vehicles and should not try any sudden, unexpected maneuvers. But why would you require them to signal every change of direction? Especially that cyclists have a right to use the full width of the lane.