Fluke has long used the venerable 2N3904 as low leakage diodes. Shorting C to B as the anode, and E as the cathode makes a fairly high current low leakage diode. If I recall correctly, you can expect reverse leakage less than 10 pA in that configuration. However, the voltage rating is around 6-7V where the connection zeners. Using this in a protection circuit, this zener action is useful. Using two 2N3904s in opposing series makes an extremely low leakage bipolar zener diode. This is used in many DMMs to protect the ohms circuitry (including the 8060A). Using just the B-E junction makes even a lower leakage diode, but very low current. You'll often see this used in sample and hold circuits where the current flow is minor, and low leakage is key. I've tested some brands of 2N3904 to less than 1 pA in this configuration. No other kind of diode can touch it. Even very expensive FET based diodes will be worse when tested over temperature. So in low voltage circuits requiring an extremely low leakage diode, 2N3904s in various connections are very useful. Best of all 2N3904s are dirt cheap. Some manufacturers outperform others, so testing more than one brand might be prudent. Photodiode effects are usually not a problem with the black epoxy coat used in most TO-92s.
Also the C+B to E connection is quite fast in turn on and turn off. Again, if you can handle the <6V voltage rating. Works great in logic circuits 5V and below.
You can also use the B-C junction as a low leakage diode. This leaks around the 20pA range (if I recall correctly, been awhile). But it has the advantage of working at higher voltage. Also, not as high current as the C+B to E configuration. But still better than any signal diode that is not FET based. Not very high speed either, but I don't recall what the speed is.
The only time I would use a FET based low leakage diode is in higher voltage circuits.