Mono Synths
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ARP

ARP Axxe

Common problems:
-Keyboard bushings
ARP

ARP Odyssey

Common problems:
-Keyboard bushings
Casio

Casio VL-1

This was my first synthesizer back in 1980, I was 7 years old, although I consider my first "real" synthesizer the Korg 01/W later in 1992. VL-Tone may look like a toy but was used by many mainstream artists from Human League to Tony Banks. German band Trio hit "Da Da Da" made a lot for the success of VL-1.

The heart of VL-1 (and several other Casio models) is a LSI chip made by Nec, the uPD1867G. Although many people think it is a analog synthesizer, it is in fact not only a digital synthesizer but also the first commercial digital synthesizer. It was released in June 1979.
Crumar

Crumar DS-2

Although the synthesizer section is monophonic, there's a paraphonic string section.
Crumar

Crumar Synthaphone

This incredible little machine has breath control and a piezo sensor to control vibrato.
I was delighted the first time I played the Synthaphone - it may look like a childrens toy but man, this is an expressive instrument. This instrument was designed in 1982 by engineer Luciano Jura of Crumar, outside his working hours as Crumar was not interested in the project. Synthaphone had a enormous success during its presentation in 1983 but only a hundred copies were sold by Crumar which makes this instrument a rare find. Synthaphone is based on the SGS M110B1 monophonic synthesizer chip and includes a Curtis CEM3320 filter.
Dave Smith Instruments

Dave Smith Instruments Evolver

Davoli

Davolisint

Very likely the first serial produced synthesizer in Europe.
Doepfer

Doepfer A100BS2

Jen

Jen SX-1000

I have two units of these little machines.
Another machine based on the M110 B1 monophonic synthesizer chip from SGS.

Korg micro-preset M-500

Moog

Moog Little Phatty

Moog

Moog Micromoog

Moog

Moog the Rogue

The Rogue uses a unusual AC 24V (~250mA) adaper with a 3.5mm mono jack. 
Moog

Moog Theremini

Novation

Novation Superbass Station

Powertran Transcendent 2000

This small monophonic synth was presented on Electronics Today International magazine as the cover article in July 1978. It was designed by Tim Orr from EMS and was offered as a DIY kit for 186.50£ plus VAT. In December 1980, ETI presented a polyphonic synthesizer accordingly named Polysynth also designed by Tim Orr.

This was the first synthesizer for some musicians, notably Thomas Dolby who retrieved one from a garbage dumpster or Joy Division guitarist Bernard Sumner who bought and assembled one.
Roland

Roland SH-2

Roland

Roland SH2000