How Low-noise  Receiving Antennas Really Work
This area deals primarily with low noise antennas, and discusses effect of antenna directivity on weak-signal reception.
Some local wintertime 350Hz BW noise, compared to a sample of signals, (on one night) was:

Noise                    -127dBm
9H1BM                 -122dBm                             5400 miles
OM0WR               -95dBm                               5100 miles
DF2PY                   -88dBm                             4600 miles
WA8OLN              -78dBm                               650 miles
W3GH                   -60dBm                              650 miles
W1AW                  -53dBm                              900 miles
W4ZV                    -32dBm                             400 miles

The above signal levels may not be typical of every night, but they show the large signal level variations between weak DX and strong one-hop signals.  This chart also shows why a direct distance-corrected multiplier does not work! Every hop adds significant attenuation, as does a signal travelling near the earth's magnetic poles.
The signal level difference between noise floor and W4ZV was 95dB. W1AW and W4ZV, both in similar directions and both with similar power, have a difference of 21 dB. This is over 100 times difference in power levels at my receiver. This illustrates how important the combination of antennas, location, and propagation (W4ZV is one sharp hop away) are, rather than power, location, distance, or antennas alone. Differences between signals from the same area can be quite pronounced.
Before talking about receiving antennas for lower frequencies, it is important to understand a few basics. We all understand the primary reason we use special receiving antenna systems is to improve signal-to-noise ratio. On the surface this sounds like the same reason we use directional transmitting antennas, but there are some very important differences between transmitting and receiving applications.

Directivity Comparison of Receiving Arrays or Antennas      
The table below rates receiving antennas in order of increasing performance. It uses directivity, with results based on noise being evenly distributed in all directions. These rankings are most accurate in the frequency range of AM broadcast, 160 or 80 meter bands when:
1.) The receiving location shows a nighttime increase in noise level. In other words, the system is not limited by local or internally generated noise, instead being limited by skywave or propagated distant noise.
2.) Thunderstorms or other local noise, such as power line noise from specific directions, does not dominate the receive system noise floor.
There will be occasional exceptions, but as a general rule the ratio of peak response in the direction of the signal to average response in all directions determines how well a receiving antenna works. In virtually all installations without clearly dominant direction or directions of noise arrival, RDF (receiving directivity factor) accurately predicts receiving antenna performance. 
RDF (directivity) will be an almost perfect indicator of what you can expect from your antenna as long as:
Noise is not from the same general direction as the desired signal
Noise field strength is not greater than the ratio of peak antenna response to depth of the pattern in the direction of noise
Noise is not coming from within the antenna's nearfield or Fresnel zone
If antennas are within two dB of each other in directivity (RDF), a lesser ranked antenna may outperform a better ranked antenna. This is because:
Direction and polarization of arriving signals and  noise constantly vary, so the relative relationship between any two individual antenna's responses will vary.
Through various unavoidable errors or omissions, antennas in the real-world may not work precisely as predicted in a model.
In a majority of cases, the following RDF (directivity) table shows relative performance of antennas in ascending order:

Antenna Type                   RDF  (dB)             20-degree forward gain (dBi)     Average Gain (dBi)
1/2λ Beverage                   4.52                                        -20.28                                                    -24.8
Vertical Omni,   w/          5.05                                        1.9                                                          -3.15
60 1/4λ radials
(Ewe Flag) Pennant         7.39                                        -36.16                                                    -43.55
K9AY                                      7.7                                          -26.23                                                    -33.93
1/2λ end-fire                     7.94                                        -20.5                                                      -28.44
1λ Beverage                       8.64                                        -14.31                                                    -22.95
Two verts optimum        9.14                                        -22.46                                                    -31.6
phasing 1/8 λ spacing
Two 1λ Beverages           10.21                                     -15.45                                                    -25.66
Echelon 1/8 λ stagger
Small 4-square 1/4 λ       10.70                                     -15.79                                                    -26.49
per side (optimum
1-1/2 λ Beverage              10.84                                     -10.88                                                    -21.72
Small 4-square 1/8λ        10.97                                     -30.28                                                    -41.52
per side (opt. phase)
Single 1.75λ  Beverage   11.16                                     -6.50                                                      -17.66
2 Broadside 1.75λ             11.36                                     -3.51                                                      -14.87
Beverages .2λ
2 Broadside 1.75λ             11.91                                     -3.50                                                      -15.41
Beverages .4λ
.625λ x .125λ spaced       12.5                                        -19.5                                                      -32.0
BS/EF vertical array
2 Broadside 1.75λ             12.98                                     -3.50                                                      -16.48
Beverages 5/8λ spacing
2 Broadside 1.75λ             13.48                                     -3.49                                                      -16.97
Beverages .75λ spacing

Gain vs. Directivity Myth
One common rumor or myth is that higher antenna gain results in improved reception. Gain is an unreliable way to predict receiving ability on frequencies below upper UHF!
A clear example is illustrated in the table above. We can follow comparisons between a single 1.75λ Beverage and various spacing pairs of 1.75λ phased Beverages. In a case where spacing is .2λ, the single Beverage has a gain of -6.5dB. A pair of Beverages spaced .2λ has a gain of -3.51dB. This is a gain increase of about 3 dB. Despite the gain increase, antenna directivity and pattern do not change a noticeable amount. RDF (directivity) only increases 0.2dB, an undetectable difference. Pattern remains essentially the same, so reception remains essentially the same. Significant new nulls, or deeper nulls, are not created at close spacing.
Gain of any spaced pair is around 3dB more than a single Beverage, but reception improves and antenna pattern changes only with relatively wide spacings. Spacing must be at least be 1/2λ or more for phased Beverages to add a reliable improvement in reception quality. Wider spacing improves null depth off the sides, and narrows front lobe beamwidth. At 3/4λ spacing directivity improvement for evenly distributed noise and QRM falls short of 3dB, although side suppression of signals improves greatly!
Of nearly equal importance, many end-fire arrays actually work better with closer spacing. For an example, compare the 1/8th wl four-square RDF with the 1/4-wl four-square array.
The antennas towards the bottom end of the chart actually do receive better.
If you ask operators who visit for contests, everyone prefers the large vertical or wide-spaced Beverage arrays. Guest operators, given a choice, almost never not use single Beverages or close-spaced Beverages.

Horizontal vs. Vertical
One popular claim is that vertically polarized antennas are noisy, and horizontally polarized antenna are quiet. Another myth related to that claim is that noise sources are predominantly vertically polarized.
There is some truth to the claim that a horizontally polarized antenna can be quieter, but this requires a special condition where the majority of noise is local extended groundwave noise.  If you use a tall vertical antenna on 160, the local extended groundwave noise is almost 20 dB stronger than the noise from a 160-meter dipole that is at 300 feet.
This is because the vertical responds better to extended groundwave than the horizontally polarized dipole. At night time, when the band opens and the dominant noise source is propagated in by skywave, there is absolutely no signal to noise advantage to the dipole!!!

DC Grounded vs. Open Antennas
Another myth is that dc grounded antennas are quieter, filtering noise by shunting it to ground. This would require the antenna to short a 1.8 MHz noise, while NOT shorting a 1.8 MHz signal!!! That is pure non-sense......Just like the myth that a G5RV can be used on 160m.