Is there a list of characters that have multiple sounds?
According to CC-CEDICT ( https://www.mdbg.net/chinese/dictionary?page=cc-cedict ) out of the 10710 simplified characters they have listed there are 777 simplified characters with different pronunciations (I'm counting pronunciation with different tones as a different pronunciation).  You could bump it up a bit by also counting the ones that are described with alternate pronunciations in the definition instead of getting a separate row in the data output (i.e. some are described in the definitions of having an alternative Taiwanese pronunciation).:

啊、挨、唉、嗳、熬、把、吧、罢、伯、膀、背、呗、奔、绷、蚌、比、贲、萹、边、摽、别、浜、屏、剥、柏、薄、捭、簸、啵、卜、哺、钸、采、操、草、嚓、叉、差、杈、衩、孱、场、沉、噌、爯、称、裎、秤、尺、冲、处、揣、创、绰、刺、囱、瘥、答、打、歺、大、待、逮、疸、担、瘅、铛、挡、档、当、倒、帱、㝵、得、澄、的、镝、坻、底、氐、地、酊、钉、斗、都、读、肚、度、垛、阿、恶、诶、嗯、发、蕃、坊、菲、分、埄、缝、夫、佛、怫、払、拂、砩、芾、服、杆、干、杠、膏、搁、葛、盖、艮、更、供、枸、勾、毂、估、莞、冠、观、桄、过、蛤、哈、厂、汗、豻、夯、好、号、镐、呵、合、纥、閤、和、喝、荷、哏、桁、横、厷、哄、汞、吽、鹄、糊、哗、华、划、化、还、晃、虺、混、豁、几、给、纪、夹、嘏、贾、假、渐、监、閒、间、将、浆、剿、矫、徼、教、筊、偈、节、结、芥、藉、解、觧、价、尽、禁、丼、劲、炅、居、车、桔、锔、柜、据、句、锯、卷、圈、悁、嶡、脚、覐、觉、角、蹶、菌、隽、楷、槛、看、扛、峇、咳、颏、轲、可、磕、克、吭、倥、焢、爌、空、佝、楛、絓、会、窾、丱、卝、傀、匮、喇、剌、啦、灠、阆、唠、落、勒、累、擂、楞、樆、离、悝、蠡、豊、鬲、丽、哩、唎、歛、敛、倞、凉、踉、量、靓、撩、了、燎、咧、淋、令、馏、咯、隆、笼、搂、喽、碌、膔、蓼、陆、露、抡、论、络、啰、偻、捋、呒、呣、蚂、吗、么、埋、蔓、谩、茆、旄、坆、闷、氓、蒙、眯、糜、谜、靡、秘、眄、缪、忞、暋、摸、抹、没、磨、脉、貉、模、姥、牟、那、呐、哪、囝、难、馕、呢、袮、伲、泥、鸟、溺、恁、拧、宁、杻、拗、弄、呶、娜、哦、嚄、熰、沤、扒、杷、耙、钯、扳、番、湴、般、板、尨、胖、刨、炮、麃、跑、泡、喷、搒、裨、劈、匹、否、辟、扁、便、匾、片、猵、漂、莩、嘌、乀、撇、拚、苹、冯、泊、酦、陂、朴、掊、仆、脯、埔、堡、瀑、铺、缉、丌、奇、荠、跂、骑、檵、稽、岂、亟、妻、卡、浅、嵌、强、抢、镪、呛、跄、蹡、蕉、悄、翘、谯、茄、切、慊、趄、厪、廑、顷、亲、嬛、璚、仇、区、焌、渠、瞿、曲、弮、犮、炔、礐、阙、雀、儿、喏、厹、女、爇、撒、塞、散、潵、丧、缲、扫、臊、刹、嗄、帹、煞、色、掺、单、扇、禅、上、杓、芍、少、稍、奓、阇、舍、参、沈、甚、乘、晟、盛、什、拾、石、莳、适、掱、橾、丨、数、刷、衰、卛、率、泷、瞤、说、厶、伺、似、俟、厕、食、圱、尿、莎、沓、踏、苔、忕、啴、弹、澹、覃、汤、樘、倘、荡、趟、叨、绦、㲈、忒、弟、鬄、佃、钿、苕、调、挑、帖、町、铤、听、侗、峒、硐、同、通、斢、头、宊、吐、囤、褪、驮、媠、拓、凹、搲、哇、呙、歪、娩、万、王、崴、硙、亹、唯、委、媁、隗、喂、为、玟、免、揾、絻、涡、薶、喔、庑、芴、巂、腊、茜、裼、纚、系、咥、扢、郄、虾、吓、厦、纤、䕭、鲜、铣、见、夅、降、相、蛸、謏、校、肖、滊、猲、劦、挟、絜、颉、莘、镡、芯、行、省、兴、宿、臭、圩、欻、砉、畜、吅、旋、削、踅、丷、亽、卪、煆、哑、压、轧、阏、埏、剡、奄、傿、咽、燕、氜、飬、铫、疟、要、耶、掖、叶、荑、尾、迆、迤、仡、佚、艾、衣、谥、殷、洇、湮、唫、龈、窨、隐、饮、巆、应、唷、哟、涌、甬、佣、繇、於、菸、俞、畲、禺、予、与、吁、忬、蔚、语、雨、圜、媛、远、约、哕、楽、乐、趯、伝、芸、晕、熨、缊、载、咱、攒、脏、藏、咋、曾、缯、喳、扎、查、楂、皻、吒、炸、侧、翟、祭、呫、粘、鳣、崭、占、长、鞝、涨、嘲、朝、着、召、折、乇、乙、蜇、辙、褶、瑱、酖、怔、挣、正、症、氏、只、乿、擿、识、鉄、中、种、重、粥、轴、朮、竺、术、属、褚、拽、转、传、篹、奘、僮、幢、戆、獞、椎、追、缀、屯、棹、蝃、濯、蓔、仔、吱、呲、訾、子、综、熜、从、卒、崒、足、赚、钻、觜、俊、嘬、琢、撮、作、酢

If you don't count pronunciation with different tones as a different pronunciation but count only those with pronunciations with either a different initial and/or final then the list drops down to 490 characters:

伯、呗、蚌、贲、浜、剥、柏、薄、捭、卜、嚓、差、孱、噌、爯、称、尺、绰、囱、瘥、歺、大、疸、铛、帱、㝵、得、澄、的、坻、底、地、都、读、度、阿、蕃、埄、佛、怫、払、拂、砩、芾、盖、蛤、厂、豻、夯、镐、呵、合、纥、閤、哏、桁、厷、汞、吽、和、鹄、还、豁、给、嘏、贾、剿、偈、芥、藉、价、丼、劲、炅、居、车、桔、枸、柜、句、嶡、脚、覐、觉、角、隽、楷、槛、扛、峇、咳、克、吭、焢、爌、佝、楛、絓、会、窾、丱、卝、傀、匮、落、勒、樆、离、悝、豊、鬲、歛、倞、靓、了、咯、碌、膔、蓼、陆、露、络、偻、捋、呒、埋、么、坆、氓、糜、谜、秘、缪、抹、没、脉、貉、模、姥、牟、囝、哪、呢、袮、鸟、溺、恁、杻、拗、弄、呶、娜、那、哦、嚄、扒、杷、耙、钯、扳、番、湴、般、板、膀、尨、胖、刨、炮、麃、搒、裨、否、辟、扁、便、匾、猵、莩、乀、拚、屏、苹、冯、泊、酦、陂、朴、脯、埔、堡、瀑、缉、丌、奇、荠、跂、骑、檵、稽、岂、亟、卡、浅、嵌、将、强、蕉、茄、慊、趄、厪、廑、亲、嬛、璚、仇、区、焌、渠、瞿、圈、弮、犮、炔、礐、雀、儿、喏、厹、女、爇、潵、缲、塞、刹、嗄、帹、色、掺、单、禅、杓、芍、奓、阇、参、沈、乘、晟、盛、什、拾、石、适、掱、橾、丨、衰、卛、率、泷、瞤、说、数、厶、伺、似、俟、厕、食、圱、尿、莎、沓、忕、啴、弹、澹、覃、汤、樘、倘、荡、叨、绦、㲈、忒、弟、鬄、佃、钿、苕、调、町、铤、侗、峒、硐、斢、宊、囤、褪、驮、媠、拓、凹、呙、娩、莞、万、崴、硙、亹、隗、玟、忞、免、絻、涡、薶、喔、恶、芴、巂、腊、茜、裼、诶、纚、系、咥、扢、郄、虾、吓、厦、纤、䕭、铣、见、夅、降、蛸、謏、校、滊、猲、劦、挟、絜、颉、解、觧、莘、镡、行、省、宿、臭、圩、欻、砉、畜、吅、削、踅、丷、亽、卪、煆、轧、阏、埏、剡、氜、飬、铫、疟、咽、叶、荑、尾、仡、佚、艾、谥、殷、洇、湮、唫、龈、听、窨、巆、涌、甬、繇、於、菸、俞、畲、禺、吁、忬、蔚、圜、约、哕、楽、乐、趯、伝、芸、熨、咱、攒、藏、咋、曾、喳、扎、查、楂、皻、侧、翟、祭、呫、粘、鳣、崭、长、鞝、嘲、朝、召、折、乇、乙、辙、褶、着、瑱、酖、氏、乿、擿、识、鉄、重、粥、朮、竺、术、属、褚、拽、转、传、篹、奘、僮、幢、戆、獞、椎、追、缀、屯、棹、蝃、濯、蓔、仔、吱、呲、综、熜、从、卒、崒、足、赚、觜、俊、嘬、琢、撮、酢