Many options. What you consider looks best is very much a matter of personal preference, and experiment will tell you more than straight instructions, but the following may help.

In this case - not the only way, in the 1st image note that the depth of field is narrow and centered around the front duck, making it and a band of ?water? sharp. Background ducks are blurred as much by focus as by rain. Rain that is seen is in the focus band. Given the depth of field, a moderately long focal length and largish aperture was probably used - there is no way to be sure but maybe 100-200 mm and f5.6 ish. Maybe. A flash can help to make rain stand out sharply (as it freezes it) but be aware that your system may tend to expose for rain exposure in the foreground, thus pushing background brightness down. Manual or semi manual exposure control will allow greater control.

In the 2nd photo they have again used depth of field to give a sharp foreground with background blur being caused both by rain and defocusing. Really high speed will freeze drops so they are separately discernible - you'll find out that they are NOT drop shaped as usually drawn but fall in a much more flattened form due to air pressure distorting the drop. Slow shutter speed gets long streaks/runs with loss of drop detail. You need to play and decide how much you want the drops to be visible versus streaming. Careful use of flash, so the photo is not dominated by it but so it tends to freeze individual drops even when overall shutter speed is slow enough to cause "runs" to show, gives a mix of both worlds. Balance with flash level control.

Note that while the "gate" photo is monochrome, the above "duck" photo may appear almost monochrome at a glance but is far from it. Background foliage has substantial green in it. Front duck has lots of brown and white and colour variation.

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Some good material here D Photo - Capturing images of rain
This describes a systematic investigation of the effect of shutter speed and aperture and more on rain images ! . Not much re rain on ground, but much else which is relevant.

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Some useful links related to rain photography:

Not all these are 100% on target but all relate to rain and photography and add some value.

23 Inspirational Images of a Rainy Day

No explanations of how here - just examples of what can be achieved. But just looking through them with other advice in mind should help. Helped me :-).

Photographing Subjects in the Rain ~~

How to photograph IN the rain

User discussion some value.

35 Beautiful Examples Of Rain Photography

Pentax forum discussion

Rainscapes photo based artworks.
Not my favourite pieces, but of possible value in the context.

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There are some useful thoughts in this PHOTOGRAPHING RAIN article - only the 4th photo down directly relates. (Nikon D50, 1/125s, f/5.6, 50 mm

but the 5th photo (bride and groom) and the associated writeup here provide some useful thoughts.

He agrees with my observation re flash (always nice to know :-).) He says:

• As for taking the first photo, keep in mind that although long shutter speeds can add a cool effect to a photograph, shorter shutter speeds can stop raindrops right in their tracks. (first article) and

• The backlight will illuminate the rain and freeze it giving you this amazing backdrop of raindrops. (second article)

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ADDED:

Len's comment got me to view the flickr version of the ducks and EXIF data is available here.
1/250th s exposure. f/4 and 500mm! The modest but not too narrow depth of field indicates that the subject was some distance away but not vastly so (no to mention the subject size). Maybe tens of metres.