Choosing a Chessmen Set
I guess every chess player has got a real chess set at home. Maybe not even one. I personally have got one but used to have two (not intentionally). I was hunting for a good used wooden chessboard for my Zagreb 1959 set (shown on the post photo). So, I bought a board with another chessmen set, which I sold afterwards. That was the standard tournament one with the German knight type.
What chess set to have is purely a matter of taste. My personal approach to chessmen choice is mostly on the practical side: The chessmen
1) should be sturdy and durable
2) should be comfortable to grab/move/play
3) should be of aesthetic shape and not to distract your attention while playing/thinking
4) should be affordable
I guess those principles are applied to choosing the sets for official tournaments. Here below I am only talking about the Staunton chessmen type.
1) Sturdy and durable.
If you noticed, chess clubs mostly use the standard plastic chess sets, which are usually non-weighted (though there are weighted versions too). They have typical L-shaped knights and are very solid, damage resistible and cheap to replace. 
Plastic is good, much stronger than wood. I would even prefer a good plastic set if I found one. There are different ones mimicking the famous chess sets like the Zurich, Marshall, Imperial, Reykjavik etc. 
But the truth is I have never seen any quality plastic sets. They are usually made very poorly: visible defects, casting seams, dangling weights etc. Also the variety of sets is very narrow. There are like basic 4-5 different types of sets with the same L-shaped knights etc. If one can recommend a good plastic set, I would be glad to check that option.
Among the wooden sets the trend is: with higher price you get elegant but more fragile chessmen. Economical to mid-range sets are more suitable for daily play and OTB analysis. Here I give you the comparison of different chessmen features and weak points.
The most vulnerable parts of chessmen are pawn collars, queen crowns, horse muzzles, tips on bishops, queens, kings heads. So, the chessmen material is usually a hard kind of wood (like boxwood etc.) to prevent them from accidental damaging.
You can clearly see that the crown on the first queen is very fragile. The black bishop's tip will certainly not survive a drop. Note, that bishops can be very sturdy even with the typical head cut (middle). The left knight's muzzle (the lower jaw) can easily break off if the piece is dropped. L-shaped knights are more vulnerable to that. Among the pawns you can notice very thin collars on A and B compared with the thick and solid ones on C and D.
One thing that slightly annoys me in my set is the sharp edge around the base (A). Once you perform takings, knocking with the sharp edge can produce dents on the taken piece (e.g. when blitzing). Other sets (like B, C, D) have the plain cylindrical base and are less likely to damage the pieces.
One can say "I'll be careful, I buy chess for playing, not for throwing the pieces". Sure, but if you analyse a lot OTB, believe me, the pieces will accidentally drop sooner or later. If you buy an expensive set you will be paranoid about all little scratches that occurred on your precious chessmen. So the concept of sturdiness is applied to the chessmen that are in regular use.
2) Comfortable and easy to grab/hold/move
Being comfortable to play means easy to grab and hold firmly while making a move. There is no much difference among the sets as all Staunton chessmen "heads" are designed for grabbing with two or three finger pinch. 
Sometimes the chessmen head shapes are slightly "off-beat" which IMO makes them uncomfortable to grab and hold (especially in blitz games). I refer mostly to knights and bishops. Some people might find that not too critical, as playing blitz we tend to grab the pieces with our whole hand rather than a pinch. 
Bishop heads A to C (and even D) are considered by me as uncomfortable. Whereas E fits the pinch firmly due to sufficient space between the head and the collar. That's why (again IMO) the Zagreb or Dubrovnik-style (E) bishop is the most comfortable.
Knights are represented by three main kinds: L-shape (the most common), the Zagreb-Dubrovnik shape (when the muzzle is faced down) and the third is all interim head positions. We will discuss the aesthetics further down. For the comfort of play the only condition is that the knights fit well in the hand and do not slip.
Most of the tournament set knights have the A profile (just flat) or B - slightly concave. Some of Zagreb sets (C) have the upper widening that prevents the knight from slipping off your fingers. The Sinquefield sets (both 2014 and 2017) have the profile as D - relatively thick stead's neck and only the place to grip on the top of the head. Yet the neck has two notches on each side for a firm pinch grip along the neck. Some of replicas might not have those adjustments that can be found in the original chess sets.
Another criterion of comfort is chessmen weight. Cheap unweighted plastic tournament sets are horrible. One touch or accidental elbow move - and the pieces are knocked over. That's why a good set has weights added. I met regular and heavy weighted pieces (like triple weighted). Heavy weights are good for blitz but so are the regular ones. I personally don't like extremely heavy pieces. Double weight is probably as far as I would go without making myself concerned about unnatural heaviness. The main thing about a good set is - the weights should be fixed/glued well and not to rattle. That's a scourge of even some quite expensive sets (due to aging), and the fixing is not easy.
The last thing is a good green cloth called felt that is glued to the bottom of chessmen. That should be a relatively thick fabric not paper. That makes pieces slide smoothly only when you move them and muffles the move sound when you lift and put them back on the board.
So that's all about the comfort. This part is relatively long as I said: comfort of play is a very important practical thing that reduces distraction.
3) Chessmen shape and aesthetic look
That will be the most disputable topic as all of us have different tastes. Out of the Staunton type chessmen I dislike the luxury ones with excessive details and ornamental work (they are very distracting and fragile) and the cheap ones with uncomfortable pieces and the pig snout knights. There should be the Golden Middle of comfort and beauty.
I want to talk about a thing that can be attributed to a wide variety of chessmen. The look of chessmen should be aesthetic or in other words visually pleasant to look at and not disturbing by any odd detail. Below are the samples of some "harmony rule violations".
a) Chessmen look disproportional. The height to base width ratio can be really weird. The FIDE guidelines state that height to base ratio should be within the range 2.0-2.5
On the above pic the Kubbel set (A) is too short but steady & practical. Set B is way too high for that basis. It's a cheap beginner set with the pieces like bowling pins: they fall over from a slightest touch. You are going to hate me, but the current WCC set is disproportional! If you compare that with the Reykjavik set (D) the difference is obvious - set (C) must be less stable. Even by the look it seems "too thin" and less aesthetic IMO.
b) Chessmen can be really ugly. That in 90% relates to knights. I will not point at any specific sets, but sometimes the knights are just too odd compared with other pieces. They can have ridiculously detailed manes of a large size or a very ugly muzzle that resembles tapir snouts. Don't the designers know that even one ugly piece can totally put people off buying the whole set? Whoever buys those must have absolute lack of taste (or not playing chess at all).
c) Chessman details. Some pieces are excessively detailed. More details make the set more expensive but at the same time it becomes more distracting and less practical. Ask the question: do you want to use the chessmen or just admire how beautiful they are? At least find the balance between the two and make your set useful.
4) Chessmen affordability.
I don't understand people buying expensive chess sets. I don't understand why "ebonised boxwood pieces" should be worse the real ebony ones. The difference in price will not make you play stronger. Add to that the constant paranoia of accidentally damaging your precious set. My current set + the used board cost me about $150. I am very happy with that set, even though I had a very strong temptation (and could afford) to buy another, more expensive one. I just didn't. Keeping the new set in the box (or on the shelf) and opening that once a month - had no practical value to me. Reasoning helped.
There are many good chess sets up to $100. There are very nice and practical sets up to $300. If you look at paying more money, in most of the cases you want the set "like player X had", or "that was used in tournament Y". That is a collector approach, not a chess player one. The only driving motive here is to possess something of aesthetic value, enjoying looking at that and feeling that in your hands. That's exactly how we finish with a bunch of things we don't really need.
Practical advice: If you want to have a good chessmen set - buy one. The common problem is that you might have got another one already. Don't buy a new one just because "the knights look much nicer". You can save that money to buy something else (chess related). If you still desire a new set - sell the old one(s). Make someone else happy with the used set(s) at a cheaper price. Otherwise, you will finish with several sets barely used for anything. Use your nice new set for analysis and play. If the set gives you inspiration that will be so much joy!
I hope that my experience can help you choose the chessmen that are suitable for you. 
