Pros:  Used PS1 racing games are generally quite affordable and easy to find via trustworthy online vendors (e.g., Amazon and eBay). And there are well over a hundred PS1 racing titles – which are compatible with all PS2 and (presumably) PS3 consoles.
Cons:  Fifth-generation-console racing games’ graphics arguably aren’t as exciting as their counterparts for more recent systems. And many PS1 racing games don’t provide “analog-thumbstick” steering (as a control option to “digital D-pad” steering).
 
Preface  (Feel free to skip ahead to my list of game titles.)
Edgar Rice Burroughs once defined golf as a mental disorder. Similarly, I myself – only partly jestingly – have long defined collecting as a species of mania. All my life I’ve been afflicted with this recurring madness. In youth I focused on philately. But in adulthood I’ve settled on books, music, and video games. I find those three subjects ultimately more gratifying than stamps, coins or cards. For not only can I passively behold my sundry objects ranged alphabetically on display shelves, but also I can open or play any single title and lose myself within a captivating alternate reality.

While my taste in music and books might seem stodgy (I collect primarily “classical” and “nonfiction”) my taste in videogaming is more wide-ranging. That said, I favor the following genres: arcade; shmups; and racing. Of the three, my favorite, by far, is racing— in virtually any form, not just the usual “cars, trucks or bikes.”
During the months when I was painstakingly building my (arguably definitive) American racing-games collection, I felt continually frustrated by the most widely disseminated, incomplete lists of “racing” titles for any of my several old consoles— including the PlayStation 2 (which is PS1-compatible). [Note:  PlayStation 3 consoles are likewise backwardly compatible with most PS1 games. However, I myself don’t presently own a PS3.]
Thus I hope my below — presumably comprehensive — list of recommendable PS1 racing-game titles may be of help to some fellow collectors of this genre [though it’s possible I’ve overlooked a title or two that merit inclusion].

However, I do have three key criteria determining (or at least influencing) which titles made my list.
First, titles like Driver or Grand Theft Auto [which I myself neither play nor collect] are primarily role-playing, not racing, games. Thus I generally don’t include such titles on this list.
Second, excessively puerile/adolescent/cartoony graphics, audio or themes (in games like Crash Team Racing and Twisted Metal) normally don’t appeal to this adult gamer. [But I’ve no such problem with certain seemingly universally beloved kart-racing titles involving Pac-Man, Kirby, Mario and Sonic that were released for later consoles and scarcely annoy adult sensibility.]

Third, if a game simply incorporates hopelessly clumsy mechanics – or glaringly ineptly rendered audio/graphics – that keep me from enjoying the overall experience, that title is anathema. [Here too, however, there’s a gray area, insofar as my below list does encompass a minority of “borderline-acceptable” titles that somehow manage to amuse me enough to merit inclusion.
Note:  I’m American. Unsurprisingly, my below list comprises only NTSC editions. And some games bear different titles (or weren’t released) in other world regions. [Consult Wikipedia for details.]
Finally, my general intention isn’t to discuss any particular game below; but the simple fact that these are the titles that made my list means each somehow merits a passing grade (though in some instances just a “D”). So, use the following list judiciously,  perhaps consulting pertinent reviews, articles or videos at the worthy Amazon, eBay, IGN, Wikipedia and YouTube to decide for yourself if a certain title belongs in your collection. 

[To see a particular game at Amazon, click its picture or title.]
1.
 
2.
 
3.
 
4.
 
5.    
 
6.
 
7.
 
8.
 
9.
 
10.
 
11.
 
12.    
 
13.
 
14.
 
15.
 
16.
 
17.
 
18.
 
19.
 
20.    
 
21.
 
22.
 
23.
 
24.    
 
25.
 
26.
 
27.
 
28.
 
29.
 
30.    
 
31.
 
32.
 
33.
 
34.
 
35.
 
36.    
 
37.      [Note:  This relatively weak game only barely qualified for this list!]

 
38.
 
39.
 
40.
 
41.    
 
42.
 
43.    
 
44.
 
45.
 
46.    
 
47.
 
48.
 
49.    
 
50.
 
51.
 
52.
 
53.       [Note:  Mine is the slightly expanded “” The (not shown below) has different cover art.]
 
54.       [Mine is the standard edition (as shown below). Also available is a (presumably slightly expanded) “.”]
 
55.
 
56.     (’96 Season)

 
57.
 
58.
 
59.
 
60.
 
61.
 
62.    
 
63.

 
64.
 
65.
 
66.
 
67.    
 
68.
 
69.
 
70.
 
71.
 
72.
 
73.
 
74.      (two-disc set, in double-wide jewel case)
 
75.
 
76.
 
77.
 
78.
 
79.
 
80.
 
81.
 
82.
 
83.
 
84.
 
85.
 
86.
 
87.
 
88.    
 
89.
 
90.
 
91.
 
92.
 
93.
 
94.
 
95.
 
96.
 
97.
 
98.     [Note:  This game — whose cutscenes overshadow its actual “racing” segments — only barely qualified for this list!]
 
99.
 
100.
 
101.
 
102.
 
103.    
 
104.
 
105.
 
106.    
 
107.
 
108.
 
109.      [Note:  Nice cover art notwithstanding, this relatively weak game only barely qualified for this list!]
 
110.
 
111.
 
112.
 
113.
 
114.
 
115.    
 
116.
 
117.
 
118.    [Note: This “original” Wipeout game is a noteworthy title for PS1 collectors; however, my take is that its gameplay’s relative shortcomings make it seem altogether superfluous for gamers who already own any of its superior sequels, including Wipeout XL and/or Wipeout 3.]
 
119.
 
120.
 
121.      [Note:  This relatively weak game only barely qualified for this list!]
 
*    *    *
You may also enjoy the following:  
The 293 Best RACING Games for PlayStation 2
The 99 Best RACING Games for Original Xbox
The 51 Best RACING Games for GameCube
The 86 Best RACING Games for Original Wii
The 135 Best RACING Games for Xbox 360