Selecting the right bridge rectifier
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.
For classic rectifiers, you want to exceed the output requirements.
In the diode datasheet, you'll see things such as forward voltage drop with respect to current and the temperature resistances of the diode. You'll also see section called absolute maximum ratings. You never want to be close to those numbers! If you want to properly select a diode, you'll read the voltage drop at the current you're interested in. Then you'll multiply the voltage by current to see the power dissipated by the diode. When you multiply the power by thermal resistance, you'll see how higher will the diode's temperature be compared to the outside temperature. This value needs to be lower than the maximum temperature of the diode or it will die. For higher currents, you can attach the diode to the heatsink in order to provide higher heat dissipation.
So let's see what happens when we take the 4 A requirement and apply it to a diode. For the example, I'll use the BYW29/200 diode which has maximum current of 8 A and voltage of 200 V. Looking at the graph in the datasheet we can see that at 4 A the forward voltage is going to be around 0.7 V. The power dissipated by the diode is going to be 2.8 W. The maximum junction temperature is +175 C. Next, we have the junction to ambient thermal resistance which is 60 C/W. So we multiply the 2.8 W by 60 C/W and get 168 C. This gives us the maximum ambient temperature of just 7 C! As we can see, the diode is pretty hard to use without any cooling.
So how do we use it? Well we add a heatsink. Another interesting note in the datasheet is the junction to case thermal resistance which is 3 C/W. To that we'll add the thermal resistance of the heatsink. In this example, I'll use one whose thermal resistance is 9.6 C/W (you may also see thermal resistance marked in K/W, but keep in mind that one degree Celsius and one kelvin are of the same magnitude). To make sure that we have good contact between the heatsink and the diode, we'll use some thermal paste too. If you have the thermal resistance of the paste, add it to the rest of thermal resistances. If you don't, just apply a "fudge factor". So we have thermal resistance of 12.6 C/W and power of 2.8 W. That gives us 35.28 C rise in junction temperature. Since I didn't have the thermal paste resistance, I'll just divide that by 0.8 and get 44.1 C. To get maximum operating temperature, we'll subtract 44.1 C from the maximum junction temperature of 175 C. This gives us maximum ambient temperature of 130.1 C, which is quite good.
Next, a little bit about the voltage. Keep in mind that your transformer's voltage will depend on the load. For unloaded transformer, it can be much higher than for transformer with full load. Next, you also have the rectified and filtered voltage. Many people here use "rules of thumb" to determine what components to use and values. Let's take a look at the 12 V example again. I'll take that 12 V to be the RMS value of the voltage of the transformer at the full load. The maximum voltage is going to be around 17 V when the transformer is at the full load. When the transformer is at no load, it could go high and 30 V to 40 V wouldn't be too surprising to see here. On the other hand, many diodes can take such voltages without any problems. For example even the simplest 1N4001 diode has maximum repetitive voltage of 50 V. The BYW29/200 from the previous example has maximum repetitive voltage of 200 V, so you don't need to worry much there. Just be sure to exceed the highest voltage you'll see in the circuit by some 20 V to 30 V and you'll be safe.
For the end, a simple method of looking for the right diode would be to just multiply the current you need by 2 and look for diodes rated at that current of higher. Then take a look at the maximum temperatures and thermal resistances to see what kind of cooling is required.