Why are there no commercial supersonic flights at present?
There a several reasons for the demise of commercial supersonic aircraft.  And there is a strong possibility we may be able to take a ride on a supersonic aircraft by 2021.
Supersonic commercial aircraft have a lot of constraints on them which are not present on subsonic commercial aircraft and which military aircraft are not subject to.

Prohibitive Development Costs
Supersonic aircraft - and in particular high supersonic and hypersonic aircraft - are considerably more expensive to design and manufacture as they require the use of novel materials and manufacturing techniques in their design.  As you pass about Mach 2, heat starts to become a major problem, weakening traditional aerospace aluminum alloys to the point that they are no longer a viable material to construct an aircraft from.  The airframes also expand and contract considerably in the heat and this must be accounted for in the design.  Designers are forced to seek out more exotic materials like Titanium alloys, high temperature composites, etc.  and it is not just structure that requires novel materials to address this - fuel systems, electrical wiring, mission systems, even sealants and paint all must be chosen so as to combat heat but still satisfy traditional design metrics like safety, durability, and economics.
The result of this are very high development costs:  Adjusted for inflation, Concorde costs the governments of England and France some $28 billion or roughly 5 1/2 times the cost that Boeing spent to develop the 747.  Similar figures were consumed on the design and development of the ill fated Boeing 2707, which never progressed past the mockup stage, even after re-design.
Limited Cabin Capacities
Super Sonic Transport (or SST) design favors a slender, needle-like fuselage which limits the number of people which the jet can carry.  The Concorde was limited to approximately 100 passenger in very narrow seats; the 2707 could carry 260 people but had a length of over 300 feet making it very awkward and unruly in existing terminal aprons or taxiways.  This also makes the SST very unprofitable for its operators, as Concorde proved.  Over the last 50 years no more than 20 SSTs have ever been operational and all with only three airlines - British Airways, Air France, and the Soviet Aeroflot - all of which were heavily subsidized by their respective governments.
Propulsion Inefficiencies Restrict Range
Design of SSTs have in the past been hampered by the high fuel consumption rates of turbojet engine imposing a design range limit of approximately 4,500 NM.  This effectively curbs sales to airlines looking for SSTs to fly the pacific rim routes where the times savings offered by a supersonic aircraft would be the most pronounced.
Supersonic Flight Over Land Still Prohibited
Supersonic aircraft generate noxious sonic booms which can cause damage to fragile structures on the ground and generally are a nuisance to the inhabitants below.  While advancements in quiet supersonic transports are happening, global aeronautical regulation bodies still prohibit supersonic flight over land, restricting an SST to subsonic flight over these regions, making its operation exponentially more expensive than a traditional subsonic aircraft.
Concerns Over Safety Operating in Rarefied Atmosphere
A high speed SST would be required to operate in the higher parts of the stratosphere in excess of 60,000 feet.  While this is achievable for military pilots in full pressure suits, there are considerable risks to civilians in street clothes at these altitudes should a rapid decompression occur.  At these heights a rapid decompression will boil your blood inside your veins (very painful way to die).  It also cannot be combated with traditional supplemental oxygen systems, making such a scenario very dangerous for passengers.
Environmental Concerns
The 'dirty burning' turbojet previously discussed create a lot of NOX emissions, far more that today's high bypass turbofans, which poses risk of increased depletion of stratospheric ozone.
Cost Savings
A conventional high bypass turbofan subsonic airliner is between 16-20 times more efficient on a cost per passenger per NM basis. A typical transatlantic ticket on a 747-8 is approximately 1500 USD.  A ticket on the Concorde was about 11,000 in today's dollars.

Taken all together and faced with the reality of designing such a craft with 3 million parts, 6 million lines of software code, all tested and validated, which is restricted only to transatlantic routes, aircraft designers quickly see the handwriting on the wall and the aircraft is doomed from the drafting paper upward.
But that's not to say the supersonic dream is dead.  Aerion in Reno, NV and Airbus are in the process of developing a 16 passenger supersonic business jet, capable of Mach 1.6, with a tentative rollout and flight in 2021.  Areion claims the jet will have a 'boomless' cruise up to Mach 1.2 at or above FL350 and hopes this will coax regulating bodies into permitting supersonic flight over land.  Whether this attracts enough customers willing to pay the estimated $100 million price tag per aircraft remains to be seen.