A lot of great options in this thread, but they're also the guns everyone has. How about something different? Don't forget if you go back far enough, S&W even made N frame 38 specials. They were made for the 38/44 round, the predecessor to the 357 magnum. There were two basic models, the fixed sight heavy duty (later the model 20), and the adj sight Outdoorsman (later the model 23).
Here is a post war transitional Outdoorsman which shipped in 1946 to Cleveland, OH.
Here is my post war transitional Heavy Duty which shipped also in 1946 to Denver, CO.
So N frame 38 specials are few and far between. What is a more practical alternative for one seeking that old world quality? Well a model 14, a K-38 target masterpiece, or a perhaps a M&P target. The M&P target was the grandfather to the modern day model 14.
Here is a 1905 4th change M&P target from probably the early to middle 1930s. SN 623XXX. Excellent fit and finish, and trigger. The sights are nice but the post war ones are better, I must admit. Also these service grips aren't the best in the hand, but the later K frame magnas work great for me. Of course K frame target grips are also compatible since M&Ps are K frame.
In this photo we have two 1905 1st change targets (top and middle - scarce) SN 106XXX and 115XXX followed by a K-38 from the early 1950s, K111XXX. S&W made great target 38s.
Just so I don't get labeled as S&W biased, here is an awesome Colt 38 special 1912 Officers model - second issue. Its not in the best shape, but its nice and original. And truthfully, for $350 - how could I lose? It actually has a 7.5in barrel which Colt still made in those days.