An explanation is necessary for this list. The BDS list of UK species seems ambiguous in that it includes pure migrants whilst I want a list of UK resident species.
I began with the 44 species of Odonata taken from Britain’s Dragonflies (Smallshire and Swash). These were the so-called “breeding species”. However, that included Sympetrum flaveolum (Yellow-winged Darter), with which I disagree; S. flaveolum has, in the past, invaded our shores on a few occasions and apparently has bred but at the time of editing  it hasn’t been seen for more than 20 years – it certainly ain’t resident IMHO.
Conversely, over recent years we have consistently seen a few newcomers in the UK. Both Lestes barbarus (Migrant Spreadwing/Southern Emerald Damselfly) and Aeshna affinis (Blue-eyed Hawker/Southern Migrant Hawker) are now established in Kent and/or southern Essex. Originally regarded as migrants, they certainly now appear to be breeding residents. Coenagrion scitulum (Dainty Damselfly), having once been a resident species that was thought to have been wiped out, was rediscovered in 2010 and is now apparently surviving on the Isle of Sheppey but is on a private site with only the county recorders apparently having access. So, these I have included in my resident list. The list also includes Anax parthenope (Lesser Emperor) as a yearly visitor rather than as a breeding resident. Thus I have ended up with my UK list of 45 species.
Species which are linked to a species page, I have seen (though not necessarily in the UK). Unlinked entries, I am yet to see. As a result of a successful though weather-plagued trip to Scotland in 2017, I am now down to one species remaining to be seen.
- Calopteryx splendens (Banded Demoiselle)
- Calopteryx virgo (Beautiful Demoiselle)
- Chalcolestes viridis (Willow Emerald Damselfly)
- Lestes sponsa (Emerald Damselfly)
- Lestes dryas (Scarce Emerald Damselfly)
- Lestes barbarus (Southern Emerald Damselfly)
- Ischnura elegans (Blue-tailed Damselfly)
- Ischnura pumilio (Scarce Blue-tailed Damselfly)
- Enallagma cyathigerum (Common Blue Damselfly)
- Coenagrion pulchellum (Variable Damselfly)
- Coenagrion puella (Azure Damselfly)
- Coenagrion mercuriale (Southern Damselfly)
- Coenagrion scitulum (Dainty Damselfly)
- Coenagrion hastulatum (Northern Damselfly) 
- Coenagrion lunulatum (Irish Damselfly) 
- Erythromma najas (Red-eyed Damselfly)
- Erythromma viridulum (Small Red-eyed Damselfly)
- Pyrrhosoma nymphula (Large Red Damselfly)
- Ceriagrion tenellum (Small Red Damselfly)
- Platycnemis pennipes (White-legged Damselfly)
- Aeshna mixta (Migrant Hawker)
- Aeshna affinis (Southern Migrant Hawker)
- Aeshna isoceles (Norfolk Hawker)
- Aeshna grandis (Brown Hawker)
- Aeshna caerulea (Azure Hawker) 
- Aeshna cyanea (Southern Hawker)
- Aeshna juncea (Common Hawker)
- Anax imperator (Emperor Dragonfly)
- Anax parthenope (Lesser Emperor Dragonfly) 
- Brachytron pratense (Hairy Dragonfly)
- Gomphus vulgatissimus (Club-tailed Dragonfly)
- Cordulegaster boltonii (Golden-ringed Dragonfly)
- Cordulia aenea (Downy Emerald)
- Somatochlora metallica (Brilliant Emerald)
- Somatochlora arctica (Northern Emerald) 
- Libellula quadrimaculata (Four-spotted Chaser)
- Libellula depressa (Broad-bodied Chaser)
- Libellula fulva (Scarce Chaser)
- Orthetrum cancellatum (Black-tailed Skimmer)
- Orthetrum coerulescens (Keeled Skimmer)
- Leucorrhinia dubia (White-faced Darter)
- Sympetrum danae (Black Darter)
- Sympetrum sanguineum (Ruddy Darter)
- Sympetrum striolatum (Common Darter)
- Sympetrum fonscolombii (Red-veined Darter) 
 – Limited to Scotland.
 – Limited to Ireland.
 – Perhaps not a breeding resident but has been recorded every year since 1998.
 – May or may not be resident but if not, it seems to re-colonize and breed every year (it can manage 2 generations a season).