The Gene Roddenberry universe is dominated by weapons, sensors and defensive systems that would generally render "fighters" as we know them useless in battle. Systems like phasers and torpedoes are both so powerful and easily-aimed as to destroy many small craft with a single shot, and so bulky the system wouldn't fit in most small craft. Downscaling most of these weapons would reduce their effectiveness, which would already be minor against a heavily-shielded capital ship. The main remaining use of a fighter as a scout ship for long-range patrols is sometimes seen, but even then ships have long-range sensors allowing a skilled operator to ascertain the approximate size and nature of an incoming force much faster than a patrol ship could be sent. Combined with the basic overhead of designing a spaceworthy ship of any size, small ships simply aren't cost-effective if you can build big.
There are several episodes of TNG and Voyager where the inadequacy of small craft against a large powerful ship is demonstrated; VOY:Dreadnought for instance had a planet's entire space force (consisting of small fighter-type craft) decimated by the Dreadnought without the missile receiving a scratch. There was a similar episode in TNG where the Enterprise crew was duped into almost attacking a pre-warp civilization with a fledgling space force. A short phaser burst to each ship did it; the Enterprise wasn't even forced off-course.
A Klingon Bird of Prey, containing only about a dozen crew and officers, would probably be the closest thing to a fighter throughout most of the canon. Though with its large wings it is similar in width to a Constitution-class starship, the actual habitable area is very small, by some accounts smaller than the Defiant-class (with a complement of about two dozen). Instead, their cloaking device makes them hard to find, which makes them perfect for patrols, scouting, deep strike, and harrassment. The Romulan Bird of Prey is similar. Both of these factions have much larger craft which would be a match or more for the Federation cruisers of the respective eras.
Jem'Hadar fighters are called that, but they're very similar in almost all respects to a Klingon Bird of Prey. The key difference being that the Jem'Hadar ships are commonly used in groups as landing craft, while Klingon ships, although landing-capable, usually stay out in space as lone wolves.
Deep Space Nine and Voyager did introduce a few Federation ships smaller than the standard cruiser-size ships of TOS and TNG. Runabouts were introduced in the late TNG series as DS9 ramped up; these were larger, armed shuttles that could stand up to small aggressors (a runabout isn't quite a match for a Jem'Hadar fighter, but if it took one by surprise it had the maneuverability to win by attrition). During Operation Return, the Federation employed attack fighters as a counter to the Jem'Hadar fighters (which were "small fries" but posed a threat in numbers to larger ships). The Ma'Quis ended up using the same ship plans, acquired from Federation defectors, to build their own fighters. The Defiant, seen in many DS9 episodes, is best described as a corvette; a small multi-crew hit-and-run ship used for patrols and deep strikes in hostile territory. The Delta Flyer would be a true fighter in most senses of the word; a heavily-armed, highly maneuverable, lightly-crewed ship somewhere between a shuttle and runabout in size.
All of these ships had some home base, either a larger cruiser or a space station. However, they were secondary to the primary purpose and armament of that ship or station. The "carrier", in terms of a ship whose main purpose is to be a floating base for these light craft, was not seen in the Federation fleet, because the primary reason they're used today is that fighters don't have the range needed to cover the globe from a stationary base, but they do have better range than most other ship armaments like artillery, torpedoes or even cruise missiles. In Star Trek, where the fighters themselves are warp-capable and relatively long-range, a ship whose primary mission is to ferry a large number of these fighters around the galaxy would just be a big target, especially when a fully-armed enemy ship can close distance at several hundred times the speed of light.
One more thing not yet mentioned is that a carrier is, first and foremost, a military weapon. Star Fleet's primary mission is exploration, with its secondary mission being peacekeeping, and ships are designed (and armed) based on that mission. Many pure "science vessels", like the Grissom, apparently weren't even armed (the Pasteur, a science/medical ship, did have some weapons but they were laughable against an enemy ship of similar size). It would be very difficult for the Federation to justify building and maintaining a massive carrier vessel, with the necessary offensive weaponry to defend itself in a battle and a fighter complement to boot, as a ship built for any other purpose than war. Even in the depths of the Dominion War, and with the Borg poking their noses in wherever they can, that's just not in Starfleet's charter. Besides, undertaking a construction project like that would likely take too long; even smaller ships like the Intrepid class take years to build, so you'd need to start construction at the first whisper of a major war, and it might launch in time to make a difference in a pivotal endgame battle, after which it does, what exactly? Patrol borders, antagonizing other factions with open displays of Federation firepower?
The main place fighters, and carriers, are seen in space sci-fi is in universes where energy shields aren't present (or have very limited effect). These universes also usually have more realistic energy generation systems than Star Trek's, and so a small craft has a short effective range and is generally not FTL-capable (unlike almost everything with a weapon in the ST universe). In such a universe, fighters now have value, as a wing of fighters can chew up larger craft, and remain effective even with horrifying attrition of their own numbers. Compounding that, larger ships' heavy guns are generally too slow to track a small, fast-moving fighter. So, it becomes a paper-rock-scissors battle, where fighter loses to corvette, which loses to capital ship, which loses to fighter. Star Wars and Babylon 5 are two well-known universes in which fighters (and ships with a permanent fighter complement) were used to great effect, but in virtually all cases the ship doing the fighter-carrying still had its own heavy guns for ship-to-ship confrontation. I can only think of one ship - the B5 Raider mothership - that wasn't combat-effective without its complement of fighters.