Across:
1 Leaves home, for example (HORSE CHESTNUT tree), and is picked up in the forest (HORSE CHESTNUT),
10 Survive (OUTLIVE) with one of the Greens (Olive = -O-LIVE) around Utah (-UT-) (OUTLIVE),
11 Legal or (legal or = ALLEGRO) dodgy (= anagram indicator) piece of lively music? (ALLEGRO),
12 Tender (“sore” = “SOAR”) reports (= homophone indicator) on the rise (SOAR),
13 Old Irish hobby, for instance (‘Irish hobby horse’ = type of Irish HORSE, now extinct), popular with all sides today (HORSE on all sides of today’s grid),
15 Local tabloids (RAGS) at the border’s race meetings (at the borders ‘race meetings’ = RAGS),
17 Detect (SEE) spark coming off speakers (‘spark’ coming off ‘speakers’ = ees = SEE),
19 One of the ranks (rank smell = ROTTEN) is corrupt (ROTTEN),
21 Put up with (ACCEPT) performance (AC-T) around Coldplay’s first (‘coldplays’ first = -C-) record (extended-play record = -EP-) (ACCEPT),
22 The Waste Land (a stretch of marshy wet land called CURRAGH) – of course, the type, that’s plain to see in Kildare? (CURRAGH plain),
23 Picture (DEPICT) cited (cited = DE-ICT) novel (= anagram indicator) about president (-P-) (DEPICT),
25 Mistakes (ERRATA) rare (rare = ERRA-) sporty (= anagram indicator) car (model -T-) with aerodynamic front (with ‘aerodynamic’ front = -A) (ERRATA),
27 Case in point – giving up nicotine (‘case in point’ – giving up nicotine/without ‘nicotine’ = asp = ASP), the poisonous type (ASP),
29 Well, it’s not (SICK) ghoulish (SICK),
30 Post (post = “stake” = “STEAK”) reports (= homophone indicator) in the butcher’s window (STEAK),
31 Have a quick look at (SCAN) couples supporting school plan (couples supporting ‘school plan’ = SCAN),
34 Rejected (= reversal indicator) most (l) evil (“evil” without “l” = evi = -IVE) book (tome = EMOT-) (EMOTIVE) that’s controversial (EMOTIVE),
35 Most of (k) the style (‘look’ without ‘k’ = LOO-) one associates with Marina (-PIER) (LOOPIER) is more eccentric (LOOPIER),
36 Teaching those nags a lesson (HORSE TRAINING) can be rewarding for those working on 22 across (HORSE TRAINING on the “Curragh”).
Down:
2 Away (OUT-) with the players (-CAST) (OUTCAST) sent to Coventry (OUTCAST),
3 Knocks back (= reversal indicator) what waiter takes home (tips = SPIT) holding kebab (SPIT),
4 Turns over (= reversal indicator) sod (sod = -DOS) for Oriental (east = -E-) church (Roman Catholic = RC = CR-) (CREDOS) with a number of doctrines (CREDOS),
5 Slip away (ELAPSE) easily (= anagram indicator) asleep (asleep = ELAPSE),
6 Presumably, holds the money for stockholders (TILL) and the banks? Time will tell! (banks ‘time will tell’ = TILL),
7 Director (-D-) in Prague (Prague = UPGRA-E) working (= anagram indicator) (UPGRADE) to get a seat in first-class coming from coach (UPGRADE),
8 Heiress hoards (heiress hoards = HORSERADISHES) pickled (= anagram indicator) root vegetables for roast beef sauce (HORSERADISHES),
9 Heard (= homophone indicator) gruff voice (“hoarse” = “horse’ = HORSE) coaches, (TRAINERS) (HORSE TRAINERS) trying to get the best out of the Arabians (Arabian horses / HORSE TRAINERS),
14 Agent (REP-) linked to Shine’s partner (-RISE and shine) (REPRISE) offering to perform again (REPRISE),
16 Graduate (M-A) rings back (= reversal indicator) Clare (CE = -EC-) and Charlie (-C-) (MECCA) from holy city (MECCA),
18 Hate leaving each other (‘hate’ leaving ‘each other’ = coher = OCHRE) in the autumnal shade (OCHRE),
20 Couldn’t get rid of cold (‘couldnt’ get rid of ‘cold’ = unt = NUT) in Brazil (NUT),
21 Held in Gulag erroneously (held in “Gulag erroneously” = AGE) for a specific number of years (AGE),
24 Funny (= anagram indicator) cool pic (cool pic = PICCOLO) of one in the pit (in orchestra pit = PICCOLO),
26 Caution (caution = AUCTION) butcher (= anagram indicator) for selling lots (AUCTION),
27 Confirm (ATTEST) a (A-) bone (-T- bone) analysis (-TEST) (ATTEST),
28 A (-A) late start (‘late’ start = -L-) to spring (leap = PAEL-) turns everything on its head (= reverse all indicator) (PAELLA) for the Spanish staple (PAELLA),
32 Some characters in Donnybrook issued (some characters in ‘Donnybrook issued’ = KISS) a welcome that’s very informal (KISS),
33 Trendiest coming out of Downing Street (‘trendiest’ coming out of ‘Downing Street’ = owng = GOWN) in formal wear (GOWN).