For Bridge Rectifier selection: Short-list parts that exceed the required maximum voltage, and the required current, by a fair margin, as described below.
For sine wave output from a transformer, the required voltage would be sqrt(2)=1.4142 times the rated transformer output voltage, as transformers are rated for RMS voltage, not peak. Also, transformers are usually, but not always, rated lower than the actual voltage they produce across the secondary with no load: This drops to the rated voltage when the transformer is carrying the rated full load current. Hence, to be on the safe side, around 2.5 times the transformer rated voltage works well for me.
For current calculation as well, 2.5 times the expected load current is healthy - since you would need the bridge to withstand the initial current surge when any reservoir capacitors following the bridge are charging up after power-on.
Now that you have the voltage and current ratings to look for, listing available parts might show you higher rated parts that are cheaper than those just meeting your requirements - so just go with the higher rated parts.
For instance, in local stores near where I live, a BR68 bridge sells for less than half of a BR36, despite the much higher rating. This is due to economies of scale - the BR68 part is just more commonly used here.
Another consideration, though, is physical size / PCB layout: Higher rated bridges tend to increase in size. Also, sometimes SIP pin-put modules are just more convenient on the PCB, compared to square pin-outs, if vertical space is not an issue.
For discrete diode selection: The same calculations apply as for the bridge. The key advantage of going with discrete parts is that heat dissipation is a bit less bothersome, since each diode has its own surrounding space to dissipate heat.
A minor additional benefit is the facility to indulge in somewhat creative PCB layouts when needed, rather than being forced to give up a specific contiguous area on the board.
Best Answer
In this application Schottky diodes will work just as well as normal diodes, by arranging them as you normally would a bridge rectifier. They are not normally used in regulators because they are more expensive and sometimes physically larger. The advantages of a Schottky diode (low forward voltage, fast switching) will not provide any benefit for an ordinary bridge rectifier.
The best option for hobby projects is to get a purpose made bridge rectifier that has all of your rectifier diodes integrated in one package. These cost about $2 in small quantities.
However, the specific part you have selected should not be used in a 120V bridge rectifier, because it will experience a peak-to-peak reverse voltage of 336V, well above the maximum rated 150V reverse voltage for this part. Don't do it.
And make sure you have a GFCI or isolation transformer between mains and your power supply.