At this point, TSA and other countries' security agencies are not permitting monopods in carry-on bags. TSA has told me that it's a stick or weapon to them. My advice, if you want to bring your monopod when you fly, is to pack it in your checked-in luggage.
Tripods, on the other hand, usually may be taken into the cabin in your carry-on, or packed in your checked-in luggage.
The choice of where to pack my tripod, has more to do with how long I'm going to be traveling, and what I need to pack in my carry-on bags, than convenience or any other factor.
When packing one's belongings, air travelers must understand airlines' rules and liability limits. In the words of US Airways, as an example,
"US Airways assumes no liability for valuable/commercial items including but not limited to: money, negotiable papers, securities, irreplaceable business documents, books, manuscripts, publications, photographic or electronic equipment, musical instruments, jewelry, silverware, precious metals, furs, antiques, artifacts, paintings and other works of art, lifesaving medication, and samples."and in addition,
"Total liability for provable direct or consequential damages resulting from the loss, delay, or damage to baggage in US Airways’ custody is limited as follows:The airlines are telling us all, don't pack anything of value, or what you can't afford to loose, in your checked-in luggage.
A. for travel wholly between U.S. points, to $3300 per customer
B. for most international travel (including domestic portions of international journeys), to $9.07 per pound ($20 per kilo) for checked baggage and $400 per customer for unchecked baggage in the custody/control of the carrier." (If your one checked-in bag weighs the maximum of 50 lbs, the maximum liability coverage offered is $453.50.)
In the US and most of the world, you're allowed a single carry-on bag, plus a personal bag. I carry a roller bag which meets the maximum size general standard of 22"x14"x9" (56cm x 36cm x 23cm). This standard does vary to an extent, but I've flown all over the world with my carry-on roller bag and it's been allowed everywhere.
My personal bag is a backpack, in particular the Lowepro CompuTrekker AW Plus, in which I take my electronics, my camera gear, and my laptop.
In my roller bag, I pack, at a minimum, any other valuables I have, my medications, toilet articles, and a complete change of clothes, including shoes or sneaks.
If I'm going to be away for a weekend, I usually only use carry-on, and can attach my tripod to the bag, or pack it inside it. It's usually attached to the outside of the bag. If I'm going away for a longer time, I take a single checked-in bag and pack my tripod in the checked-in bag. When I pack the tripod in the checked-in luggage, I remove its ball head which I pack in my back-pack.
While I haven't had the problem myself, I have heard from other photographers who travel with their tripods that every once in a while, they have been forced to check-in their tripod when they tried to carry it on the plane. It becomes a target for theft at that point, and if not in a padded bag may be seriously damaged. Ask to "gate check" the tripod at that point, as it will be less likely to be stolen that way.
So, to enjoy taking those nighttime or sunrise/sunset photos, and others, with your tripod or monopod, I suggest, if at all possible, to put it in your checked-in luggage.