#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));
        }
    }
}