Which preposition should be used when referring to an exact date?

Solution 1:

A variation of your first sentence:

My test is on the 22nd of June.

Although you may write on June 22, 2013, in a conversation I would say on the 22nd of June - notice the the and the nd ending of the 22nd. Alternatively you can say on June 22nd.

More here In, At, On + Time or Date

• in + month or year - in March, in 2013
• on + date (with the year or without it) or day of the week - on April 2, on March 3, 2014, on Saturday
• at + clock time, midnight, noon - at 3:30 p.m., at 4:01, at noon

Remember also...

• in + season - in the summer, in the winter
• in + morning, afternoon, evening - in the morning, in the evening
• at + night - at night