When singing a melody from sight, go through the following process before attempting to sing it.
First, identify the meter. The meter can be found by looking at the top number of the time signature. Then identify the beat value and the division value from looking at the bottom number. (Revisit Meter and time signatures if necessary.)
Next, identify the key. The key signature will give you two possibilities for key: one major and one minor. Then look at the first and last notes of the melody: these will often be tonic, especially the last note (this is not always true for excerpts of longer melodies, though). This will help you decide between the two possibilities presented by the key signature. Lastly, look for any accidentals. Raised pitches that would be la or ti in the minor key suggested by the signature will confirm that minor key as the key of the melody.
Then scan the rhythm for the fastest notes, and the pitches for the hardest passages. Choose a tempo that will make these passages comfortable.
Find the highest and lowest notes. When singing by yourself, you can sing in any key you like. Choose a key that will put the highest and lowest notes in your range.
Once you have key and tempo chosen, orient yourself to the key. Use the solfège patterns on pp. 51 and 89 of the Karpinski Manual.
After orienting yourself to the key, look for the hardest and trickiest passages. Sing through a couple of these licks out loud, perhaps adding or removing notes in order to figure them out, then sing them as written.
Be sure to sing the lowest and highest notes out loud before you begin the melody. This way you will know both the sound and the feel of those notes ahead of time, which will help you perform them and detect errors as you sing.
Finally, with a comfortable key and tempo, and at least a rough handle on the toughest licks, sing the melody straight through, while conducting the meter, without stopping.