When do the hands of the clock line up?

When do they form equal angles? When do they point in opposite directions?

The hour & minute hands align perfectly every 43200/11 seconds (1 hour, 5 minutes, 27.273 seconds). This happens 11 times every 12 hours.
Exactly in between the times they align perfectly, they are at exact opposition (11 times every 12 hours).
The hour & second hands align perfectly every 43200/719 seconds (60.083 seconds). This happens 719 times every 12 hours.
Exactly in between the times they align perfectly, they are at exact opposition (719 times every 12 hours).
The minute & second hands align perfectly every 3600/59 seconds (61.017 seconds). This happens 708 times every 12 hours.
Exactly in between the times they align perfectly, they are at exact opposition (708 times every 12 hours).
The hour hand bisects the angle between the minute & second hands every 4320/73 seconds (59.178 seconds). This happens 730 times every 12 hours.
The minute hand bisects the angle between the hour & second hands every 43200/697 seconds (61.980 seconds). This happens 697 times every 12 hours.
The second hand bisects the angle between the hour & minute hands every 43200/1427 seconds (30.273 seconds). This happens 1427 times every 12 hours.

The last three data files also include the angle that is made by the bisecting hand. We also consider it a bisection when the bisector splits the "back" of the angle (i.e, splits the > 180 degree angle side).

Twelve o'clock is included in the counts of 1, 3, 5, 7, 8, 9. Six o'clock is included in the counts of 2, 4, 5, 7. The rest of the times are all unique, giving 5723 total "interesting" times on the clock.

You may also consider "interesting" the times when two clock hands form a 90-degree angle. This happens exactly in between when a pair of hands align and when they oppose, so it happens 22 times for H&M, 1438 times for H&S, and 1416 times for M&S. Note that three o'clock and nine o'clock are counted multiple times, so this brings the total interesting times to 8595.

How did you calculate these values?

The math for this is fairly elementary. First of all, let's make the following observations:

The second hand moves 6 degrees every second.
The minute hand moves 1/10 degree every second.
The hour hand moves 1/120 degree every second.

So if t seconds have elapsed since 12 o'clock, the hands have moved 6t, t/10, and t/120 degrees, respectively. We will need to interpret these degrees mod 360.

Now, let's say we want to find the times where the hour hand (H) is exactly between the minute (M) and second (S) hands. This means that the angular distance from M to H is the same as the distance from H to S. In other words:

M - H = H - S (mod 360)

Simplifying this, we get that M-2H+S must be a multiple of 360. If we substitute in S=6t, M=t/10, and H=t/120, we get that:

t/10 - t/60 + 6t = (73/12)t is a multiple of 360

In other words, t must be a multiple of 360*(12/73) = 4320/73. So the hour hand bisects the other two every 4320/73 = 59.178 seconds, which is exactly what happens.

The other calculations for when one hand is exactly between two other hands are very similar and follow the same structure.

If we want to find out when two hands line up (say, the hour and second hands), this is even easier. The two hands aligning simply means that

H = S (mod 360)

Or in other words, S-H is a multiple of 360. Substituting in, this means that

6t - t/120 = (719/120)t is a multiple of 360

So the alignment happens every 360*(120/719) = 43200/719 = 60.083 seconds, which matches what we calculated.