Modern military forces focus on using sensors to identify the target, then sending fairly high precision weaponry to deal with the problem. Since Dragons are usually depicted as being subsonic, it is highly unlikely they can outrun missiles, and opposition fighters will probably launch several at a time (either a volley from one fighter, or several fighters firing from different angles) to overwhelm any fire/acid/cold etc. breath weapons. Heat seeking and radar guided missiles can also manoeuvre at more than 9"G", so out turning an oncoming missile is going to be problematic as well.
Ground based surface to air missiles have similar properties, so coming into the air defense umbrella isn't going to be very healthy either.
Ground combat will be equally problematic. Unless the dragon is fighting 1950 era Communist Chinese troops advancing in a Zerg rush, submachine guns blazing, it is also going to be in trouble.
Artillery can strike at distances of 40km using guided shells like "Excalibur". A 155 shell will have a considerable amount of kinetic energy, so even a strike by a dud shell will be devastating, and a direct hit by a 155mm Excalibur is considered sufficient to destroy a modern tank. All the gunners need is a good fire mission from a well sited forward observer, UAV or trained infantryman hiding in the woods and death will descend from the sky.
Of course, if you don't have Excalibur, then filling a grid square with HE can have a similar effect, either batteries of cannon firing traditional shells or multiple rocket launchers can rapidly fill a 1 X 1 kilometre square with enough fire and steel to disrupt or destroy a modern mechanized formation, so any dragon in the square is also in for a world of hurt.
Moving closer, the dragon will be subject to direct fire weapons like ATGM's launched from helicopters or ground mounts (effective ranges measured from 10+k to several hundred metres [the point where the missile is stabilized after launch and can assume a flight profile towards the target]), tanks (APDS-FS) rounds are effective against enemy tanks to @ 3 km) or even rapid fire automatic cannon from infantry IFV's at 2km range. And the dragon isn't going against these systems one on one, military forces move and fight as combined arms teams, so the dragon is being hit by a multitude of rounds ranging from direct fire from tanks and hand held ATGM's (like Javelin), indirect fire like mortar rounds and even infantry machine guns (.50 cal is pretty dangerous at 2km, but even 7.62mm medium or general purpose machine guns have effective ranges of 800m +).
About the only thing the Dragon would not find dangerous is the dismounted infantryman armed with a 5.56 assault rifle or light machine gun. The soldier could give a good account of himself if he has something like an AT-4 or RPG for close range anti tank use.
At sea, much the same thing will happen, just different weapons systems will be in play (the modern USN is very close to adding 100kW laser weapons into the mix as well).
So short answer is the dragon or dragons will be relentlessly hunted down by sensor units, then targeted by the longest range weapons that can be brought to bear.