Thank you to Gurney. Definitions are underlined in the clues.
Across
1. Love, Nablus exotic, mysterious, unlike this puzzle, one hopes! (10)
UNSOLVABLE : Anagram of(… exotic) LOVE, NABLUS.
7. Fish limit getting around resistance (4)
CARP : CAP(to limit/to place a cap/a maximum on) containing(getting around) R(symbol for “electrical resistance” in physics).
A variety: 
9. Just starting, find useful Japanese index showing mountain name (4)
FUJI : 1st letters, respectively, of(Just starting) “find useful Japanese index“.
10. Cunning, pen in unusual entries (10)
SERPENTINE : PEN contained in(in) anagram of(unusual) ENTRIES.
Defn: …/treacherous.
11. Something wrong with a peninsula (6)
CRIMEA : CRIME(something wrong/an offence) plus(with) A.
Defn: … that juts into the Black Sea.
12. Legacy perhaps of dictator’s attitude to come into view (8)
HEIRLOOM : Homophone of(dictator’s) “air”(attitude/a distinctive manner or appearance) + LOOM(to come into view/to appear as a vague form).
Defn: A valuable object that has belonged to a family for generations, an example/perhaps, of a legacy.
13. Vessels, modified, foil all threats initially (8)
FLOTILLA : Anagram of(modified) [FOIL ALL + 1st letter of(… initially) “threats” ].
Defn: A fleet of …/ships.
15. Joke about northern group (4)
GANG : GAG(a joke/a quip) containing(about) N(abbrev. for “northern”).
17. Willing to consider musical work, European, on periphery (4)
OPEN : OP(abbrev. for “opus”, a major musical work) + 1st and last letters of(…, on periphery) “European“.
19. Attribute of required standard, extremely trusty (8)
PROPERTY : PROPER(of required standard/conforming to social standards or conventions) + 1st and last letters of(extremely) “trusty“.
Defn: An …/a characteristic of something.
22. Antagonise a learner – that is needless primarily, battle regularly lost (8)
ALIENATE : A + L(letter displayed by a learner driver) + IE(abbrev. for “id est”/that is) + 1st letter of(… primarily) “needless” + “battle” minus its 1st, 3rd and 5th letters(regularly lost).
23. Perhaps Mercury‘s scheme, out of this world? (6)
PLANET : PLAN(a scheme/a plot) + ET(abbrev. for “extra-terrestrial”/alien/out of this world).
Defn: An example of which/perhaps is Mercury.
25. Van meal bit off? Maybe yes, maybe no (10)
AMBIVALENT : Anagram of(… off) VAN MEAL BIT.
Defn: …/sitting on the fence/unwilling to take sides about something or someone.
26. Comfort from herbal medicine (4)
BALM : Hidden in(from) “herbal medicine“.
27. Wizard entertaining millions – a high point (4)
ACME : ACE(a wizard/an expert in a particular field or activity) containing(entertaining) M(abbrev. for “millions”).
28. Compete, given award we’re told for initiative (10)
ENTERPRISE : ENTER(to compete/to participate in a competition) plus(given) homophone of(… we’re told) “prize”(an award for, say, winning a competition).
Down
2. New country group, based on test ignoring India, seen as indifferent (7)
NEUTRAL : N(abbrev. for “new”) + EU(abbrev. for the European Union, a group of many countries of Europe) plus(based on, in a down clue) “trial”(a test/a try-out) minus(ignoring) “I”(abbrev. for “India”).
3. Dislike arising from leader leaving speaking platform (5)
ODIUM : 1st letter of(leader) deleted from(leaving) “podium”(a speaking platform/a raised stand on which a speaker stands).
4. In a picture, six upset America and supporter (8)
VISUALLY : VI(Roman numeral for “six”) + reversal of(upset, in a down clue) US(abbrev. for things American) plus(and) ALLY(a supporter/someone taking your side).
5. Annual transfer of allotted space, we hear – Doris maybe kind of tense (8,7)
BIRTHDAY PRESENT : Homophone of(…, we hear) “berth”(an allotted space for a vessel to dock) + DAY(an example of whom is/maybe, Doris Day, American singer) + PRESENT(a kind of/one of the tenses in grammar).
Defn: A transfer of something/a gift from one to another celebrating his/her birthday.
6. Feature writer, outspoken, supported by Liberal papers (6)
EYELID : Homophone of(…, outspoken) “I”(the writer using the self-referential pronoun) placed above(supported by, in a down clue) [ L(abbrev. for a member of the Liberal Party) + ID(abbrev. for “identity papers”).
Defn: … on one’s face.
7. Go at a clue in different way – making a list (9)
CATALOGUE : Anagram of(… in different way) GO AT A CLUE.
8. Neglected animal track – sad (3-4)
RUN-DOWN : RUN(a track regularly used by a particular animal, say, a rabbit) + DOWN(sad/down in the dumps).
14. Strange generation, losing love for fruit (9)
TANGERINE : Anagram of(Strange) [ “generation” minus(losing) “O”(letter denoting 0/love in tennis scores).
16. Device, electronic say, provides regular traveller with parking for month (8)
COMPUTER : “commuter”(a regular traveller of some distance to work) with “p”(abbrev. for “parking”) replacing(with … for …) “m”(abbrev. for “month”).
18. European note Conservative finds controversial (7)
POLEMIC : POLE(a European, specifically a national of Poland) + MI(a note on the music sol-fa scale) + C(abbrev. for a member of the Conservative Party).
20. Have an impact, about right – one’s garden feature (7)
TRELLIS : TELL(to have an impact/to be effective) containing(about) R(abbrev. for “right”) + I(Roman numeral for “one”)‘S .
21. Pledge to include artist’s style of music (6)
GARAGE : GAGE(a pledge/a valuable object deposited as a guarantee of good faith) containing(to include) RA(post-nominals for a Member of the Royal Academy of Arts, an artist).
Defn: A genre of electronic dance music originating in the early 1990s.
24. A doctor with little hesitation identifying warning colour (5)
AMBER : A + MB(abbrev. for Medicinae Baccalaureus/Bachelor of Medicine, a doctor) plus(with) ER(a short/little word expressing one’s hesitation when speaking).
Defn: …, as seen in traffic lights, to warn of the impending change to red.
An entertaining grid from Gurney with crisp cluing throughout.
Liked 1a, 10, 5d and 6d best.
Thanks to Gurney and to Scchua for a most thoroughly explained and illustrated blog. Just a small typo in 11a which should read CRIME not CRIM.
Very satisfying and not too taxing. I usually need help parsing one or two, but this one was very straightforward. The only thing I didn’t know was that a gage is a pledge. I thought it was a fruit.
Everything in steadily for this until I reached POLEMIC, which I didn’t know could be an adjective (rather than a noun) and finally GARAGE which I guessed from the def, never having heard of GAGE for ‘pledge’. We’ve had CRIMEA a bit recently – I doubt it’s coincidental.
Favourite was UNSOLVABLE, which I parsed as a double def + wordplay.
Thanks v. much to scchua and Gurney
I too was hesitant about “GARAGE”, otherwise all solved and parsed. Thanks to both.
All pretty straightforward except that I missed Eyelid. Should have persevered. Thanks to all.
Thanks Diane. Typo corrected.
Thanks for the blog, GAGE was in a puzzle very recently or I would not have got that meaning, it is where we get mortgage from.
A lot of good , neat clues here, SERPENTINE and HEIRLOOM were very nice.
Thanks Gurney for a pleasant solve over breakfast. I used a word finder to get EYELID and I couldn’t fully parse GARAGE but otherwise it was smooth sailing. Favourites were UNSOLVABLE, HEIRLOOM, and COMPUTER. Thanks scchua for your illustrated blog.
Many thanks, scchua, for the excellent blog and many thanks also to all who commented.
An enjoyable and certainly not 1ac puzzle; just the thing for our mid-afternoon break. We particularly liked FLOTILLA and POLEMIC among several others.
Thanks, Gurney and scchua.
Thanks Gurney and scchua
A rare single sitting solve after work yesterday, taking only a minute or two longer than the Monday puzzle. Still lots to enjoy throughout – especially liked the penny drops with the COMMUTER / COMPUTER and BIRTHDAY PRESENT. Thought that ‘something wrong’ for CRIME, ENTER for ‘compete’, ‘country group’ for EU and ‘annual transfer’ for BIRTHDAY PRESENT were all clever definitions.
Ended up in the NW corner with NEUTRAL and CRIMEA the last couple in.
5dn is a dreadful clue. ’Annual transfer’ = Birthday Present ?
Since when has a present been a transfer?
FFS! …and you have to be old enough to have heard of Doris Day!
Berlinda@12 re 5dn: Chambers 2014 gives “the passing of property or some other right to another person (law)” as a definition of transfer n. It does not seem too much of stretch to me to take this out of the legal context.