I personally agree with most of what La'ra says.
I would like to start off by defining what "civilian" actually means to me in this context. I mean
non-government owned and operated vessels, which pretty much rules out
only Federation diplomatic ships, Federation official bureaucratic ships (Office of the Interplanetary Affairs, Federation One, Interstellar Parking Commisson, etc), and Starfleet (police/military) ships.
As such, I think that civilian ships in the Federation will be as common as roaches in a slum. Corporations will still exist in the future. Corporations such as computer manufacturers, shipping companies, romantic cruises to stunningly beautiful stellar and planetary sites of natural beauty, luxury goods importers on most well-established planets, etc. etc.
The good ol' UF of P may not be purely capitalistic any more by the 24th century, but it will still have a supply and demand, free-enterprise market setup. Otherwise, how can it maintain its economic growth and stability? Someone wants something they cannot get where they are? Someone will ship it to them. If it is legal, and has mass market appeal, it will be shipped in bulk. And since this is bulk shipping on an interstellar scale, supplying the planetary market, and sometimes on a loop run to several planets, space bulk-carriers are going to be multi-multi-multi-megatonne monstrosities. Flying Boxcars writ cosmic.
Similarly, regular passenger services between planets--and in a free, democratic, even if socialist utopic society, people are free to go where they want, when they want. Massive interstellar liners will carry thousands or tens of thousands of passengers in cyclic runs between the major planets, also carrying low-priority mail, cargo, and luggage. Colonial movers fit into this too.
Because these will be (relatively) slow ships, they'll have to have endurance, lots of distractions for the passengers, and all sorts of special facilities to cater to all sorts of alien life support needs and wants.
If you have limited time, you can charter a small ship, a 10-passenger courier or something, and go straight from A to B. Similarly, colonial movers can fit here too, a slightly larger cargo ship chartered by a family or group of families.
If the item you want or the place you go to is illegal or restricted, there are always adventurous Human souls, never mind aliens, who will sneak it in or you there for a reward. There are private pleasure yachts (a ship that cannot or isn't allowed to carry cargo or paying passengers), and independent "jacks of all trades" who will take any job they can find--like Cyrano Jones, like the guy McCoy tries to charter to take him to Genesis, like the small merchantman that Valkris pays to meet with Kruge, like Kassidy Yates who runs her own ship, etc. etc.
Basically, it all boils down to the fact that Starfleet is
not going to be the mass passenger service for civilians. And civilians in a free society
travel. For pleasure, for business, for work, on assignment, in emergencies, you name it.
Roll into that the non-aligned worlds and other stellar nations like the Klingons, and you have international, cross border travel, too.
Therefore, even in a UFP of only 150 members, each planet having a population of around 3 billion each, that's 450 billion people, and doesn't take into account the myriad outposts, colonies, protectorates, planetary science missions, etc. etc. Even if only a thousandth of a percent on each UFP "member" planet travel annually, thats 4.5 million people hauling themselves across the stars.
Government-regulated freighters? You gotta be kidding me.
As for firepower... There is always the Orion Syndicate and the Ferengi who will sell anyone what they want and find a way too get what is wanted to sell. The number of civilian ships that can stand up to a Starfleet cruiser will be low, but there will be some. I belive that the vast majority of armed, independent ships will be on the level of the SFC Orion CR compared to the F-CA, technologially. It is frigate-type vessel, armed and shielded accordingly. Enough of them swarming, or a couple of lucky captains can at least force the retreat of a Navy cruiser. Also, they can ambush and lay traps.
Also lay into this the legitimate, non-aligned world governments who could be as developed as the Feds but just want nothing to to with the Feds. They'll do a brisk business in the arms trade, and there's nothing the Feds can do about it beyond, perhaps, economic sanctions, which would just piss of the non-aligned worlds and make them anti-Fed. The Orions themselves fall into this category.
Bottom line: Hundreds of thousands of civilian Federation ships of all types.
Interstellar Corporations will have almost as much resources as Starfleet, though it will be illegal to arm their ships beyond a certain self-defence level, and prison-worthy to field a private 'security force'.
Federation-sanctioned security fleets will be
incredibly closely monitored, if they are allowed to exist at all.
Citizen-owned and operated Federation ships will have to have all the correct permits and permissions to be armed at all, and will have much lesser armaments (if any) and power than Starfleet ships, with a possible very few exceptions. Warp and shield technology is centuries old by this point, and will be commercially available, if costly for the private owner. Warp engines are apparently relatively simple to build and maintain, it is their
fuel which is closely regulated. It
will be a huge expense to own and operate a ship, but it already is in our present world.
Non-aligned ships operating in Federation space will be able to see off Federation police ships and smaller Starfleet patrol ships, but will be scared off by the Starfleet big boys unless there's a huge advantage they can play.
This is how I see the Federation, in either the 23rd or the 24th century. And beyond. Humans are curious. They want to "go see" for themselves.