They can climb vertically, but this works best if they are several tons below their maximum take-off mass.
Fighter jet engines need a lot of fuel, and at the beginning of the flight the aircraft will be too heavy for vertical climb.
Also, the landing gear would need to be rearranged if the plane is to take off from any airport.
Even a thrust/weight ratio slightly above 1 at maximum take-off mass will not be enough, because the aircraft needs some airspeed for its control surfaces to become effective.
If no thrust vectoring is installed, the aircraft will be uncontrollable in its initial ascent.
The Harrier VTOL jet uses bleed air which is ducted to nozzles at the extreme ends of fuselage and wing for low-speed attitude control.
It is conceivable that the fighter will hang vertically on a wall, with its wheels locked in clutches which will release it when the needed thrust is reached.
With thrust vector control the aircraft could be controlled over the full trajectory until it transits to horizontal flight, and could even land vertically.
But this would need specially prepared airfields and use much more fuel than a conventional take-off, leaving less fuel for the mission.
aviation.stackexchange.com/users/1961/Peter Kämpf
Yes, they can accelerate straight up (even at max weight in some cases), but to accelerate straight up from 0 airspeed requires some kind of control to keep the aircraft stable.
All of the aircraft's normal control surfaces only work with air flowing across them, so if you stood it up and pushed the throttles forward it would simply tip itself over.
This is why VTOL aircraft always have more than one point creating thrust, they use thrust to stabilize the aircraft as it rises.
The other consideration (as Peter mentioned) is that it's less efficient to climb this way, meaning less fuel for the mission, lower takeoff weight, or some other tradeoff.
aviation.stackexchange.com/users/7752/Joel M.
The closest the US ever got to it is the ZEL program for the F-100 Super Sabre.
Basically, put the F-100 on a cruise missile launch rail, strap one hell of a big rocket booster to its arse, and run for cover.
Used to have a 90 minute VHS tape about the project, they got so far as to design underground launch ramps with nuclear blast proof doors to launch the fighters after Soviet bombers.
Never went operational though, by the time it had progressed to where it could work in practice
the Nike SAM system was mature enough the ZEL program was no longer needed.
aviation.stackexchange.com/users/900/jwenting
Well they sometimes do.
A pilot or flight lead can request an unrestricted climb to cruising altitude for the purposes of practicing intercepts or other training maneuvers as this F-22 Raptor demonstrates.
But this does employ a conventional takeoff and accelerating the jet to a pre-determined speed before going vertical.
In regards to an engine thrust greater than 1:1, this applies to static conditions at sea level, generally using full afterburning.
As you increase in altitude, the rated thrust that the engines can produce degrades with decreasing air density.
The idea of a pure VTOL fighter has been explored with the "tail sitter" designs such as the Convair XFY-1 Pogo or Lockheed's entry .
These designs were feasible for a proposed cruiser based fighter interceptor, but the configuration made it difficult to land and the project was abandoned shortly after a series of test flights.
Both the Harrier series of attack aircraft and the new F-35B can take off and land vertically, but at a greatly reduced fuel an dpayload required for hovering; These aircraft prefer short take-off operations from ship with a warlord, then land vertically once this is depleted with completion of a mission.
aviation.stackexchange.com/users/15637/Carlo Felicione
One other factor to consider: if a fighter with a thrust to weight ratio of greater than one can launch vertically off of a ship, how does that fighter land back on the ship?
Being out in the ocean, there's no where else to land, and if the ship has a flight deck for landing, may as well use it for a much safer conventional takeoff with a much higher combat load.
This was the issue faced with the three tail sitters the US financed in the 1950's, from Convair, Lockheed, and Ryan.
The bottom line: landing on the tail, with the pilot looking over their shoulder in the attempt, turned out to be far too difficult, on stationary land, with a huge landing pad, and no battle damage.
Trying to land a tail sitter on a moving ship after carrying out a combat mission would have been even more difficult.
All three efforts were abandoned as being impractical in military conditions.
aviation.stackexchange.com/users/22436/tj1000
I've heard the claim repeated many times that some modern fighter jets have an installed thrust greater than their weight, so theoretically they could accelerate straight up.  I've never actually heard of this being done, however. It sounds useful, though, especially on the deck of an aircraft carrier. Imagine: we could take out the whole catapult system, and replace it with some sort of a system to pick up an aircraft and turn it vertical.  Why is this not done?
Why don't fighter jets take off vertically?
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<takeoff><fighter><jet><aircraft-carrier><vtol>
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aviation.stackexchange.com/users/3170/raptortech97
Fighter jets can take off vertically but with a reduced payload. Also vertical take-off requires some kind of stabilizer.  The Harrier and F-35B usually take off non-vertically and then land vertically as the reduced fuel reduces the payload. There have been designs for VTOL fighter jets. There are also existing aircraft that are able to take off and land vertically but they prefer short take-offs from a ship. Moreover, fighter jets are actually able to climb vertically but they will be too heavy at the start of a flight.  
Fighter jets are able to climb vertically but they will be too heavy at the start of a flight. A vertical climb is most successful when the jets are significantly below their maximum take-off weight.  Vertical acceleration from 0 airspeed requires control to ensure stability. There have been designs for VTOL fighter jets. There are also existing aircraft that are able to take off and land vertically but they prefer short take-offs from a ship.