Chromosomes become visible during prophase, the stage of mitosis during which the nuclear envelope disappears and the chromosomes shorten and condense. After prophase, chromosomes remain visible until mitosis completes.
Prophase is an involved process and while the chromosomes are visible throughout, they look different depending on the stage of the process the cell is in. During early prophase, the chromosomes are working to condense and shorten in preparation for cell division and they appear under the microscope as a condensed mass; it is hard to distinguish individual chromosomes at this point. However, by the time prometaphase is about to begin, the chromosomes have condensed tightly enough that they can be readily distinguished, even by the untrained eye.
The fact that chromosomes become visible during prophase and remain so throughout mitosis has practical medical implications that extend beyond satisfying the curiosity of scientists. Doctors commonly call for a karyotype, a composite image of a person's chromosomes, when they suspect that a newborn child has genetic abnormalities. For example, since Down's syndrome is caused by an extra copy of chromosome 21, a doctor can easily diagnose it with a karyotype that reveals that extra chromosome visually. Since chromosomes are most visible directly after prophase, cytogeneticists in search of visible chromosomes begin the process of karyotyping by arresting cells during metaphase of mitosis.