In the last 15 years or so I always used one of these 1602 LCD modules if I wanted some kind of text display in a µC project.
They are dirt cheap, robust and have a relatively low pin count (4bit mode). At least the newer versions (white on blue) have a pretty good contrast and readability.
However, they are somewhat slow on the display side (fast changing values are barely readable), the communication needs bit-banging (no HW/DMA solution like for SPI) and they are not very flexible (only 8 programmable characters or so).
Last but not least, for some bizarre reason, the connector is (nearly) always on top. So while you can mount the LCD directly to a PCB with a right angle header for a front plate display, it will than be bottom up.
Anyway, for my next project, 16x2 characters seem a bit limiting, so I'm in the market for a new default display with good availability (also for the years to come), low price (<10€, the lower the better), low pin count (SPI preferred) and easy mounting option (ideally, a pin header to mount to the PCB with a right angle header). E.g. a bare panel with flat connector is not what I'm after. I don't really need color, just a good contrast. Resolution can be low (like 128x64), for higher resolutions, the display must have its own display buffer.
Now if you need more than 4 lines, the next obvious choice would be a 1604 or 2004, but this quite an increase in price and they share all the issues I have with the 1602 - including the bottom up problem.
For only a bit more you also get a 12864 (128x64) graphic LCD which has a nice size and doesn't have the bottom up issue. Monochrome display and the low resolution is perfectly fine as you don't have much RAM for a screenbuffer in a small µC anyway.
Then again, it's the same old technology as the 1602 with the same bit banging low level interface. Plus the whole programming (two rows because of two controllers etc.) is a bit clumsy.
Next option is a Nokia 5110/3310 display. They are cheap, easy to mount and support SPI.
Yet they are very small and have a somewhat low contrast. Last but not least, they are not produced any more and refurbished from used Nokias. So it's questionable if you still can buy them in a few years.
OLED displays are very bright, come in nice (low) resolutions and usually support SPI, but the affordable ones are too small for my purposes. Besides they tend to burn in (-> need to invert the display now and then) and have a limited lifespan.
Finally, there are 2,2" SPI 320x240 LCDs with ILI9340/ILI9341 controllers. They are available for less than a 12864 (~6€), which is rather intriguing. The controller is a bit complex, but there are libraries and a simple bitmap/font blitting should be doable anyway. My main issue is that the connector is on the side, so for PCB mouting, you either need to mount it vertically or use a small PCB as some kind of riser board.
Currently, I tend to use the latter (320x240 SPI LCD) option even though I wouldn't use either the 256k colors nor the SD card on its back. Still, from a mounting point if view, I'd prefer the 12864 monochrome display.
Any options I overlooked? What are you using?