#C#, 62 bytes#

n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");

Full program and verification:

using System;
namespace OutputOrdinalNumbers
{
 class Program
 {
 static void Main(string[] args)
 {
 Func<int,string>f= n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");
 
 for (int i=1; i<=124; i++)
 Console.WriteLine(f(i));
 }
 }
}

C#, 62 bytes

n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");

Full program and verification:

using System;
namespace OutputOrdinalNumbers
{
 class Program
 {
 static void Main(string[] args)
 {
 Func<int,string>f= n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");
 
 for (int i=1; i<=124; i++)
 Console.WriteLine(f(i));
 }
 }
}

#C#, 62 bytes#

n=>n+""+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");

Full program and verification:

using System;
namespace OutputOrdinalNumbers
{
 class Program
 {
 static void Main(string[] args)
 {
 Func<int,string>f= n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");
 
 for (int i=1; i<=124; i++)
 Console.WriteLine(f(i));
 }
 }
}

#C#, 62 bytes#

n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");

Full program and verification:

using System;
namespace OutputOrdinalNumbers
{
 class Program
 {
 static void Main(string[] args)
 {
 Func<int,string>f= n=>n+(n/10%10==1||(n%=10)<1||n>3?"th":n<2?"st":n<3?"nd":"rd");
 
 for (int i=1; i<=124; i++)
 Console.WriteLine(f(i));
 }
 }
}